Calculus Examples

$y = {(- x^{2} + 6 x - 5)}^{3}$

Step 1

Simplify ${(- x^{2} + 6 x - 5)}^{3}$ .

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Step 1.1

Use the Multinomial Theorem.

$y = {(- x^{2})}^{3} + 3 {(- x^{2})}^{2} (6 x) + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2

Simplify terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Apply the product rule to $- x^{2}$ .

$y = {(- 1)}^{3} {(x^{2})}^{3} + 3 {(- x^{2})}^{2} (6 x) + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.2

Raise $- 1$ to the power of $3$ .

$y = - {(x^{2})}^{3} + 3 {(- x^{2})}^{2} (6 x) + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.3

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 1.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = - x^{2 \cdot 3} + 3 {(- x^{2})}^{2} (6 x) + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.3.2

Multiply $2$ by $3$ .

$y = - x^{6} + 3 {(- x^{2})}^{2} (6 x) + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.4

Rewrite using the commutative property of multiplication.

$y = - x^{6} + 3 \cdot 6 {(- x^{2})}^{2} x + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.5

Multiply $3$ by $6$ .

$y = - x^{6} + 18 {(- x^{2})}^{2} x + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.6

Apply the product rule to $- x^{2}$ .

$y = - x^{6} + 18 ({(- 1)}^{2} {(x^{2})}^{2}) x + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.7

Raise $- 1$ to the power of $2$ .

$y = - x^{6} + 18 (1 {(x^{2})}^{2}) x + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.8

Multiply ${(x^{2})}^{2}$ by $1$ .

$y = - x^{6} + 18 {(x^{2})}^{2} x + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.9

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.2.1.9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = - x^{6} + 18 x^{2 \cdot 2} x + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.9.2

Multiply $2$ by $2$ .

$y = - x^{6} + 18 x^{4} x + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.10

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 1.2.1.10.1

Move $x$ .

$y = - x^{6} + 18 (x \cdot x^{4}) + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.10.2

Multiply $x$ by $x^{4}$ .

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Step 1.2.1.10.2.1

Raise $x$ to the power of $1$ .

$y = - x^{6} + 18 (x^{1} x^{4}) + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = - x^{6} + 18 x^{1 + 4} + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.10.3

Add $1$ and $4$ .

$y = - x^{6} + 18 x^{5} + 3 (- x^{2}) {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.11

Multiply $- 1$ by $3$ .

$y = - x^{6} + 18 x^{5} - 3 x^{2} {(6 x)}^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.12

Apply the product rule to $6 x$ .

$y = - x^{6} + 18 x^{5} - 3 x^{2} (6^{2} x^{2}) + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.13

Rewrite using the commutative property of multiplication.

$y = - x^{6} + 18 x^{5} - 3 \cdot 6^{2} x^{2} x^{2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.14

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.2.1.14.1

Move $x^{2}$ .

$y = - x^{6} + 18 x^{5} - 3 \cdot 6^{2} (x^{2} x^{2}) + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.14.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = - x^{6} + 18 x^{5} - 3 \cdot 6^{2} x^{2 + 2} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.14.3

Add $2$ and $2$ .

$y = - x^{6} + 18 x^{5} - 3 \cdot 6^{2} x^{4} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.15

Raise $6$ to the power of $2$ .

$y = - x^{6} + 18 x^{5} - 3 \cdot 36 x^{4} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.16

Multiply $- 3$ by $36$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + {(6 x)}^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.17

Apply the product rule to $6 x$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 6^{3} x^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.18

Raise $6$ to the power of $3$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} + 3 {(- x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.19

Apply the product rule to $- x^{2}$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} + 3 ({(- 1)}^{2} {(x^{2})}^{2}) \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.20

Raise $- 1$ to the power of $2$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} + 3 (1 {(x^{2})}^{2}) \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.21

Multiply ${(x^{2})}^{2}$ by $1$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} + 3 {(x^{2})}^{2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.22

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.2.1.22.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} + 3 x^{2 \cdot 2} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.22.2

Multiply $2$ by $2$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} + 3 x^{4} \cdot - 5 + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.23

Multiply $- 5$ by $3$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 6 (- x^{2}) (6 x) \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.24

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.2.1.24.1

Move $x$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 6 (- (x \cdot x^{2})) \cdot 6 \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.24.2

Multiply $x$ by $x^{2}$ .

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Step 1.2.1.24.2.1

Raise $x$ to the power of $1$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 6 (- (x^{1} x^{2})) \cdot 6 \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.24.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 6 (- x^{1 + 2}) \cdot 6 \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.24.3

Add $1$ and $2$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 6 (- x^{3}) \cdot 6 \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.25

Multiply $- 1$ by $6$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} - 6 x^{3} \cdot 6 \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.26

Multiply $6$ by $- 6$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} - 36 x^{3} \cdot - 5 + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.27

Multiply $- 5$ by $- 36$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 180 x^{3} + 3 {(6 x)}^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.28

Apply the product rule to $6 x$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 180 x^{3} + 3 (6^{2} x^{2}) \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.29

Raise $6$ to the power of $2$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 180 x^{3} + 3 (36 x^{2}) \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.30

Multiply $36$ by $3$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 180 x^{3} + 108 x^{2} \cdot - 5 + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.31

Multiply $- 5$ by $108$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 180 x^{3} - 540 x^{2} + 3 (- x^{2}) {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.32

Multiply $- 1$ by $3$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 180 x^{3} - 540 x^{2} - 3 x^{2} {(- 5)}^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.33

Raise $- 5$ to the power of $2$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 180 x^{3} - 540 x^{2} - 3 x^{2} \cdot 25 + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.34

Multiply $25$ by $- 3$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 180 x^{3} - 540 x^{2} - 75 x^{2} + 3 (6 x) {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.35

Multiply $6$ by $3$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 180 x^{3} - 540 x^{2} - 75 x^{2} + 18 x {(- 5)}^{2} + {(- 5)}^{3}$

Step 1.2.1.36

Raise $- 5$ to the power of $2$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 180 x^{3} - 540 x^{2} - 75 x^{2} + 18 x \cdot 25 + {(- 5)}^{3}$

Step 1.2.1.37

Multiply $25$ by $18$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 180 x^{3} - 540 x^{2} - 75 x^{2} + 450 x + {(- 5)}^{3}$

Step 1.2.1.38

Raise $- 5$ to the power of $3$ .

$y = - x^{6} + 18 x^{5} - 108 x^{4} + 216 x^{3} - 15 x^{4} + 180 x^{3} - 540 x^{2} - 75 x^{2} + 450 x - 125$

Step 1.2.2

Simplify by adding terms.

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Step 1.2.2.1

Subtract $15 x^{4}$ from $- 108 x^{4}$ .

$y = - x^{6} + 18 x^{5} - 123 x^{4} + 216 x^{3} + 180 x^{3} - 540 x^{2} - 75 x^{2} + 450 x - 125$

Step 1.2.2.2

Add $216 x^{3}$ and $180 x^{3}$ .

$y = - x^{6} + 18 x^{5} - 123 x^{4} + 396 x^{3} - 540 x^{2} - 75 x^{2} + 450 x - 125$

Step 1.2.2.3

Subtract $75 x^{2}$ from $- 540 x^{2}$ .

$y = - x^{6} + 18 x^{5} - 123 x^{4} + 396 x^{3} - 615 x^{2} + 450 x - 125$

Step 2

Set $y$ as a function of $x$ .

$f (x) = - x^{6} + 18 x^{5} - 123 x^{4} + 396 x^{3} - 615 x^{2} + 450 x - 125$

Step 3

Find the derivative.

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Step 3.1

By the Sum Rule, the derivative of $- x^{6} + 18 x^{5} - 123 x^{4} + 396 x^{3} - 615 x^{2} + 450 x - 125$ with respect to $x$ is $\frac{d}{d x} [- x^{6}] + \frac{d}{d x} [18 x^{5}] + \frac{d}{d x} [- 123 x^{4}] + \frac{d}{d x} [396 x^{3}] + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$ .

$\frac{d}{d x} [- x^{6}] + \frac{d}{d x} [18 x^{5}] + \frac{d}{d x} [- 123 x^{4}] + \frac{d}{d x} [396 x^{3}] + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.2

Evaluate $\frac{d}{d x} [- x^{6}]$ .

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Step 3.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{6}$ with respect to $x$ is $- \frac{d}{d x} [x^{6}]$ .

$- \frac{d}{d x} [x^{6}] + \frac{d}{d x} [18 x^{5}] + \frac{d}{d x} [- 123 x^{4}] + \frac{d}{d x} [396 x^{3}] + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$- (6 x^{5}) + \frac{d}{d x} [18 x^{5}] + \frac{d}{d x} [- 123 x^{4}] + \frac{d}{d x} [396 x^{3}] + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.2.3

Multiply $6$ by $- 1$ .

$- 6 x^{5} + \frac{d}{d x} [18 x^{5}] + \frac{d}{d x} [- 123 x^{4}] + \frac{d}{d x} [396 x^{3}] + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.3

Evaluate $\frac{d}{d x} [18 x^{5}]$ .

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Step 3.3.1

Since $18$ is constant with respect to $x$ , the derivative of $18 x^{5}$ with respect to $x$ is $18 \frac{d}{d x} [x^{5}]$ .

$- 6 x^{5} + 18 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 123 x^{4}] + \frac{d}{d x} [396 x^{3}] + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$- 6 x^{5} + 18 (5 x^{4}) + \frac{d}{d x} [- 123 x^{4}] + \frac{d}{d x} [396 x^{3}] + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.3.3

Multiply $5$ by $18$ .

$- 6 x^{5} + 90 x^{4} + \frac{d}{d x} [- 123 x^{4}] + \frac{d}{d x} [396 x^{3}] + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.4

Evaluate $\frac{d}{d x} [- 123 x^{4}]$ .

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Step 3.4.1

Since $- 123$ is constant with respect to $x$ , the derivative of $- 123 x^{4}$ with respect to $x$ is $- 123 \frac{d}{d x} [x^{4}]$ .

$- 6 x^{5} + 90 x^{4} - 123 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [396 x^{3}] + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 6 x^{5} + 90 x^{4} - 123 (4 x^{3}) + \frac{d}{d x} [396 x^{3}] + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.4.3

Multiply $4$ by $- 123$ .

$- 6 x^{5} + 90 x^{4} - 492 x^{3} + \frac{d}{d x} [396 x^{3}] + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.5

Evaluate $\frac{d}{d x} [396 x^{3}]$ .

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Step 3.5.1

Since $396$ is constant with respect to $x$ , the derivative of $396 x^{3}$ with respect to $x$ is $396 \frac{d}{d x} [x^{3}]$ .

$- 6 x^{5} + 90 x^{4} - 492 x^{3} + 396 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$- 6 x^{5} + 90 x^{4} - 492 x^{3} + 396 (3 x^{2}) + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.5.3

Multiply $3$ by $396$ .

$- 6 x^{5} + 90 x^{4} - 492 x^{3} + 1188 x^{2} + \frac{d}{d x} [- 615 x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.6

Evaluate $\frac{d}{d x} [- 615 x^{2}]$ .

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Step 3.6.1

Since $- 615$ is constant with respect to $x$ , the derivative of $- 615 x^{2}$ with respect to $x$ is $- 615 \frac{d}{d x} [x^{2}]$ .

$- 6 x^{5} + 90 x^{4} - 492 x^{3} + 1188 x^{2} - 615 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.6.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 6 x^{5} + 90 x^{4} - 492 x^{3} + 1188 x^{2} - 615 (2 x) + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.6.3

Multiply $2$ by $- 615$ .

$- 6 x^{5} + 90 x^{4} - 492 x^{3} + 1188 x^{2} - 1230 x + \frac{d}{d x} [450 x] + \frac{d}{d x} [- 125]$

Step 3.7

Evaluate $\frac{d}{d x} [450 x]$ .

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Step 3.7.1

Since $450$ is constant with respect to $x$ , the derivative of $450 x$ with respect to $x$ is $450 \frac{d}{d x} [x]$ .

$- 6 x^{5} + 90 x^{4} - 492 x^{3} + 1188 x^{2} - 1230 x + 450 \frac{d}{d x} [x] + \frac{d}{d x} [- 125]$

Step 3.7.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 6 x^{5} + 90 x^{4} - 492 x^{3} + 1188 x^{2} - 1230 x + 450 \cdot 1 + \frac{d}{d x} [- 125]$

Step 3.7.3

Multiply $450$ by $1$ .

$- 6 x^{5} + 90 x^{4} - 492 x^{3} + 1188 x^{2} - 1230 x + 450 + \frac{d}{d x} [- 125]$

Step 3.8

Differentiate using the Constant Rule.

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Step 3.8.1

Since $- 125$ is constant with respect to $x$ , the derivative of $- 125$ with respect to $x$ is $0$ .

$- 6 x^{5} + 90 x^{4} - 492 x^{3} + 1188 x^{2} - 1230 x + 450 + 0$

Step 3.8.2

Add $- 6 x^{5} + 90 x^{4} - 492 x^{3} + 1188 x^{2} - 1230 x + 450$ and $0$ .

$- 6 x^{5} + 90 x^{4} - 492 x^{3} + 1188 x^{2} - 1230 x + 450$

Step 4

Set the derivative equal to $0$ then solve the equation $- 6 x^{5} + 90 x^{4} - 492 x^{3} + 1188 x^{2} - 1230 x + 450 = 0$ .

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Step 4.1

Factor the left side of the equation.

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Step 4.1.1

Factor $6$ out of $- 6 x^{5} + 90 x^{4} - 492 x^{3} + 1188 x^{2} - 1230 x + 450$ .

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Step 4.1.1.1

Factor $6$ out of $- 6 x^{5}$ .

$6 (- x^{5}) + 90 x^{4} - 492 x^{3} + 1188 x^{2} - 1230 x + 450 = 0$

Step 4.1.1.2

Factor $6$ out of $90 x^{4}$ .

$6 (- x^{5}) + 6 (15 x^{4}) - 492 x^{3} + 1188 x^{2} - 1230 x + 450 = 0$

Step 4.1.1.3

Factor $6$ out of $- 492 x^{3}$ .

$6 (- x^{5}) + 6 (15 x^{4}) + 6 (- 82 x^{3}) + 1188 x^{2} - 1230 x + 450 = 0$

Step 4.1.1.4

Factor $6$ out of $1188 x^{2}$ .

$6 (- x^{5}) + 6 (15 x^{4}) + 6 (- 82 x^{3}) + 6 (198 x^{2}) - 1230 x + 450 = 0$

Step 4.1.1.5

Factor $6$ out of $- 1230 x$ .

$6 (- x^{5}) + 6 (15 x^{4}) + 6 (- 82 x^{3}) + 6 (198 x^{2}) + 6 (- 205 x) + 450 = 0$

Step 4.1.1.6

Factor $6$ out of $450$ .

$6 (- x^{5}) + 6 (15 x^{4}) + 6 (- 82 x^{3}) + 6 (198 x^{2}) + 6 (- 205 x) + 6 (75) = 0$

Step 4.1.1.7

Factor $6$ out of $6 (- x^{5}) + 6 (15 x^{4})$ .

$6 (- x^{5} + 15 x^{4}) + 6 (- 82 x^{3}) + 6 (198 x^{2}) + 6 (- 205 x) + 6 (75) = 0$

Step 4.1.1.8

Factor $6$ out of $6 (- x^{5} + 15 x^{4}) + 6 (- 82 x^{3})$ .

$6 (- x^{5} + 15 x^{4} - 82 x^{3}) + 6 (198 x^{2}) + 6 (- 205 x) + 6 (75) = 0$

Step 4.1.1.9

Factor $6$ out of $6 (- x^{5} + 15 x^{4} - 82 x^{3}) + 6 (198 x^{2})$ .

$6 (- x^{5} + 15 x^{4} - 82 x^{3} + 198 x^{2}) + 6 (- 205 x) + 6 (75) = 0$

Step 4.1.1.10

Factor $6$ out of $6 (- x^{5} + 15 x^{4} - 82 x^{3} + 198 x^{2}) + 6 (- 205 x)$ .

$6 (- x^{5} + 15 x^{4} - 82 x^{3} + 198 x^{2} - 205 x) + 6 (75) = 0$

Step 4.1.1.11

Factor $6$ out of $6 (- x^{5} + 15 x^{4} - 82 x^{3} + 198 x^{2} - 205 x) + 6 (75)$ .

$6 (- x^{5} + 15 x^{4} - 82 x^{3} + 198 x^{2} - 205 x + 75) = 0$

Step 4.1.2

Factor $- x^{5} + 15 x^{4} - 82 x^{3} + 198 x^{2} - 205 x + 75$ using the rational roots test.

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Step 4.1.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 75, \pm 3, \pm 25, \pm 5, \pm 15$

$q = \pm 1$

Step 4.1.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 75, \pm 3, \pm 25, \pm 5, \pm 15$

Step 4.1.2.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 4.1.2.3.1

Substitute $1$ into the polynomial.

$- 1^{5} + 15 \cdot 1^{4} - 82 \cdot 1^{3} + 198 \cdot 1^{2} - 205 \cdot 1 + 75$

Step 4.1.2.3.2

Raise $1$ to the power of $5$ .

$- 1 \cdot 1 + 15 \cdot 1^{4} - 82 \cdot 1^{3} + 198 \cdot 1^{2} - 205 \cdot 1 + 75$

Step 4.1.2.3.3

Multiply $- 1$ by $1$ .

$- 1 + 15 \cdot 1^{4} - 82 \cdot 1^{3} + 198 \cdot 1^{2} - 205 \cdot 1 + 75$

Step 4.1.2.3.4

Raise $1$ to the power of $4$ .

$- 1 + 15 \cdot 1 - 82 \cdot 1^{3} + 198 \cdot 1^{2} - 205 \cdot 1 + 75$

Step 4.1.2.3.5

Multiply $15$ by $1$ .

$- 1 + 15 - 82 \cdot 1^{3} + 198 \cdot 1^{2} - 205 \cdot 1 + 75$

Step 4.1.2.3.6

Add $- 1$ and $15$ .

$14 - 82 \cdot 1^{3} + 198 \cdot 1^{2} - 205 \cdot 1 + 75$

Step 4.1.2.3.7

Raise $1$ to the power of $3$ .

$14 - 82 \cdot 1 + 198 \cdot 1^{2} - 205 \cdot 1 + 75$

Step 4.1.2.3.8

Multiply $- 82$ by $1$ .

$14 - 82 + 198 \cdot 1^{2} - 205 \cdot 1 + 75$

Step 4.1.2.3.9

Subtract $82$ from $14$ .

$- 68 + 198 \cdot 1^{2} - 205 \cdot 1 + 75$

Step 4.1.2.3.10

Raise $1$ to the power of $2$ .

$- 68 + 198 \cdot 1 - 205 \cdot 1 + 75$

Step 4.1.2.3.11

Multiply $198$ by $1$ .

$- 68 + 198 - 205 \cdot 1 + 75$

Step 4.1.2.3.12

Add $- 68$ and $198$ .

$130 - 205 \cdot 1 + 75$

Step 4.1.2.3.13

Multiply $- 205$ by $1$ .

$130 - 205 + 75$

Step 4.1.2.3.14

Subtract $205$ from $130$ .

$- 75 + 75$

Step 4.1.2.3.15

Add $- 75$ and $75$ .

$0$

Step 4.1.2.4

Since $1$ is a known root, divide the polynomial by $x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- x^{5} + 15 x^{4} - 82 x^{3} + 198 x^{2} - 205 x + 75}{x - 1}$

Step 4.1.2.5

Divide $- x^{5} + 15 x^{4} - 82 x^{3} + 198 x^{2} - 205 x + 75$ by $x - 1$ .

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Step 4.1.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

Step 4.1.2.5.2

Divide the highest order term in the dividend $- x^{5}$ by the highest order term in divisor $x$ .

				-	$x^{4}$
	$x$	-	$1$	-	$x^{5}$	+	$15 x^{4}$	-	$82 x^{3}$	+	$198 x^{2}$	-	$205 x$	+	$75$

Step 4.1.2.5.3

Multiply the new quotient term by the divisor.

$x^{4}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

Step 4.1.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{5} + x^{4}$

$x^{4}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

Step 4.1.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

Step 4.1.2.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{4}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

Step 4.1.2.5.7

Divide the highest order term in the dividend $14 x^{4}$ by the highest order term in divisor $x$ .

$x^{4}$

$14 x^{3}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

Step 4.1.2.5.8

Multiply the new quotient term by the divisor.

$x^{4}$

$14 x^{3}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

Step 4.1.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $14 x^{4} - 14 x^{3}$

$x^{4}$

$14 x^{3}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

Step 4.1.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$14 x^{3}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

Step 4.1.2.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{4}$

$14 x^{3}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

Step 4.1.2.5.12

Divide the highest order term in the dividend $- 68 x^{3}$ by the highest order term in divisor $x$ .

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

Step 4.1.2.5.13

Multiply the new quotient term by the divisor.

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

Step 4.1.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 68 x^{3} + 68 x^{2}$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

Step 4.1.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

$130 x^{2}$

Step 4.1.2.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

$130 x^{2}$

$205 x$

Step 4.1.2.5.17

Divide the highest order term in the dividend $130 x^{2}$ by the highest order term in divisor $x$ .

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

$130 x^{2}$

$205 x$

Step 4.1.2.5.18

Multiply the new quotient term by the divisor.

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

$130 x^{2}$

$205 x$

$130 x^{2}$

$130 x$

Step 4.1.2.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $130 x^{2} - 130 x$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

$130 x^{2}$

$205 x$

$130 x^{2}$

$130 x$

Step 4.1.2.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

$130 x^{2}$

$205 x$

$130 x^{2}$

$130 x$

$75 x$

Step 4.1.2.5.21

Pull the next terms from the original dividend down into the current dividend.

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

$130 x^{2}$

$205 x$

$130 x^{2}$

$130 x$

$75 x$

$75$

Step 4.1.2.5.22

Divide the highest order term in the dividend $- 75 x$ by the highest order term in divisor $x$ .

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

$130 x^{2}$

$205 x$

$130 x^{2}$

$130 x$

$75 x$

$75$

Step 4.1.2.5.23

Multiply the new quotient term by the divisor.

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

$130 x^{2}$

$205 x$

$130 x^{2}$

$130 x$

$75 x$

$75$

$75 x$

$75$

Step 4.1.2.5.24

The expression needs to be subtracted from the dividend, so change all the signs in $- 75 x + 75$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

$130 x^{2}$

$205 x$

$130 x^{2}$

$130 x$

$75 x$

$75$

$75 x$

$75$

Step 4.1.2.5.25

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x$

$1$

$x^{5}$

$15 x^{4}$

$82 x^{3}$

$198 x^{2}$

$205 x$

$75$

$x^{5}$

$x^{4}$

$14 x^{4}$

$82 x^{3}$

$14 x^{4}$

$14 x^{3}$

$68 x^{3}$

$198 x^{2}$

$68 x^{3}$

$68 x^{2}$

$130 x^{2}$

$205 x$

$130 x^{2}$

$130 x$

$75 x$

$75$

$75 x$

$75$

$0$

Step 4.1.2.5.26

Since the remander is $0$ , the final answer is the quotient.

$- x^{4} + 14 x^{3} - 68 x^{2} + 130 x - 75$

Step 4.1.2.6

Write $- x^{5} + 15 x^{4} - 82 x^{3} + 198 x^{2} - 205 x + 75$ as a set of factors.

$6 ((x - 1) (- x^{4} + 14 x^{3} - 68 x^{2} + 130 x - 75)) = 0$

Step 4.1.3

Factor $- x^{4} + 14 x^{3} - 68 x^{2} + 130 x - 75$ using the rational roots test.

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Step 4.1.3.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 75, \pm 3, \pm 25, \pm 5, \pm 15$

$q = \pm 1$

Step 4.1.3.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 75, \pm 3, \pm 25, \pm 5, \pm 15$

Step 4.1.3.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 4.1.3.3.1

Substitute $1$ into the polynomial.

$- 1^{4} + 14 \cdot 1^{3} - 68 \cdot 1^{2} + 130 \cdot 1 - 75$

Step 4.1.3.3.2

Raise $1$ to the power of $4$ .

$- 1 \cdot 1 + 14 \cdot 1^{3} - 68 \cdot 1^{2} + 130 \cdot 1 - 75$

Step 4.1.3.3.3

Multiply $- 1$ by $1$ .

$- 1 + 14 \cdot 1^{3} - 68 \cdot 1^{2} + 130 \cdot 1 - 75$

Step 4.1.3.3.4

Raise $1$ to the power of $3$ .

$- 1 + 14 \cdot 1 - 68 \cdot 1^{2} + 130 \cdot 1 - 75$

Step 4.1.3.3.5

Multiply $14$ by $1$ .

$- 1 + 14 - 68 \cdot 1^{2} + 130 \cdot 1 - 75$

Step 4.1.3.3.6

Add $- 1$ and $14$ .

$13 - 68 \cdot 1^{2} + 130 \cdot 1 - 75$

Step 4.1.3.3.7

Raise $1$ to the power of $2$ .

$13 - 68 \cdot 1 + 130 \cdot 1 - 75$

Step 4.1.3.3.8

Multiply $- 68$ by $1$ .

$13 - 68 + 130 \cdot 1 - 75$

Step 4.1.3.3.9

Subtract $68$ from $13$ .

$- 55 + 130 \cdot 1 - 75$

Step 4.1.3.3.10

Multiply $130$ by $1$ .

$- 55 + 130 - 75$

Step 4.1.3.3.11

Add $- 55$ and $130$ .

$75 - 75$

Step 4.1.3.3.12

Subtract $75$ from $75$ .

$0$

Step 4.1.3.4

Since $1$ is a known root, divide the polynomial by $x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- x^{4} + 14 x^{3} - 68 x^{2} + 130 x - 75}{x - 1}$

Step 4.1.3.5

Divide $- x^{4} + 14 x^{3} - 68 x^{2} + 130 x - 75$ by $x - 1$ .

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Step 4.1.3.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

Step 4.1.3.5.2

Divide the highest order term in the dividend $- x^{4}$ by the highest order term in divisor $x$ .

				-	$x^{3}$
	$x$	-	$1$	-	$x^{4}$	+	$14 x^{3}$	-	$68 x^{2}$	+	$130 x$	-	$75$

Step 4.1.3.5.3

Multiply the new quotient term by the divisor.

			-	$x^{3}$
$x$	-	$1$	-	$x^{4}$	+	$14 x^{3}$	-	$68 x^{2}$	+	$130 x$	-	$75$
			-	$x^{4}$	+	$x^{3}$

Step 4.1.3.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{4} + x^{3}$

			-	$x^{3}$
$x$	-	$1$	-	$x^{4}$	+	$14 x^{3}$	-	$68 x^{2}$	+	$130 x$	-	$75$
			+	$x^{4}$	-	$x^{3}$

Step 4.1.3.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

Step 4.1.3.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

Step 4.1.3.5.7

Divide the highest order term in the dividend $13 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$13 x^{2}$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

Step 4.1.3.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$13 x^{2}$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

Step 4.1.3.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $13 x^{3} - 13 x^{2}$

$x^{3}$

$13 x^{2}$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

Step 4.1.3.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$13 x^{2}$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

$55 x^{2}$

Step 4.1.3.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$13 x^{2}$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

$55 x^{2}$

$130 x$

Step 4.1.3.5.12

Divide the highest order term in the dividend $- 55 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$13 x^{2}$

$55 x$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

$55 x^{2}$

$130 x$

Step 4.1.3.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$13 x^{2}$

$55 x$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

$55 x^{2}$

$130 x$

$55 x^{2}$

$55 x$

Step 4.1.3.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 55 x^{2} + 55 x$

$x^{3}$

$13 x^{2}$

$55 x$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

$55 x^{2}$

$130 x$

$55 x^{2}$

$55 x$

Step 4.1.3.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$13 x^{2}$

$55 x$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

$55 x^{2}$

$130 x$

$55 x^{2}$

$55 x$

$75 x$

Step 4.1.3.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$13 x^{2}$

$55 x$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

$55 x^{2}$

$130 x$

$55 x^{2}$

$55 x$

$75 x$

$75$

Step 4.1.3.5.17

Divide the highest order term in the dividend $75 x$ by the highest order term in divisor $x$ .

$x^{3}$

$13 x^{2}$

$55 x$

$75$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

$55 x^{2}$

$130 x$

$55 x^{2}$

$55 x$

$75 x$

$75$

Step 4.1.3.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$13 x^{2}$

$55 x$

$75$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

$55 x^{2}$

$130 x$

$55 x^{2}$

$55 x$

$75 x$

$75$

$75 x$

$75$

Step 4.1.3.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $75 x - 75$

$x^{3}$

$13 x^{2}$

$55 x$

$75$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

$55 x^{2}$

$130 x$

$55 x^{2}$

$55 x$

$75 x$

$75$

$75 x$

$75$

Step 4.1.3.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$13 x^{2}$

$55 x$

$75$

$x$

$1$

$x^{4}$

$14 x^{3}$

$68 x^{2}$

$130 x$

$75$

$x^{4}$

$x^{3}$

$13 x^{3}$

$68 x^{2}$

$13 x^{3}$

$13 x^{2}$

$55 x^{2}$

$130 x$

$55 x^{2}$

$55 x$

$75 x$

$75$

$75 x$

$75$

$0$

Step 4.1.3.5.21

Since the remander is $0$ , the final answer is the quotient.

$- x^{3} + 13 x^{2} - 55 x + 75$

Step 4.1.3.6

Write $- x^{4} + 14 x^{3} - 68 x^{2} + 130 x - 75$ as a set of factors.

$6 ((x - 1) ((x - 1) (- x^{3} + 13 x^{2} - 55 x + 75))) = 0$

Step 4.1.4

Factor $- x^{3} + 13 x^{2} - 55 x + 75$ using the rational roots test.

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Step 4.1.4.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 75, \pm 3, \pm 25, \pm 5, \pm 15$

$q = \pm 1$

Step 4.1.4.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 75, \pm 3, \pm 25, \pm 5, \pm 15$

Step 4.1.4.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 4.1.4.3.1

Substitute $3$ into the polynomial.

$- 3^{3} + 13 \cdot 3^{2} - 55 \cdot 3 + 75$

Step 4.1.4.3.2

Raise $3$ to the power of $3$ .

$- 1 \cdot 27 + 13 \cdot 3^{2} - 55 \cdot 3 + 75$

Step 4.1.4.3.3

Multiply $- 1$ by $27$ .

$- 27 + 13 \cdot 3^{2} - 55 \cdot 3 + 75$

Step 4.1.4.3.4

Raise $3$ to the power of $2$ .

$- 27 + 13 \cdot 9 - 55 \cdot 3 + 75$

Step 4.1.4.3.5

Multiply $13$ by $9$ .

$- 27 + 117 - 55 \cdot 3 + 75$

Step 4.1.4.3.6

Add $- 27$ and $117$ .

$90 - 55 \cdot 3 + 75$

Step 4.1.4.3.7

Multiply $- 55$ by $3$ .

$90 - 165 + 75$

Step 4.1.4.3.8

Subtract $165$ from $90$ .

$- 75 + 75$

Step 4.1.4.3.9

Add $- 75$ and $75$ .

$0$

Step 4.1.4.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- x^{3} + 13 x^{2} - 55 x + 75}{x - 3}$

Step 4.1.4.5

Divide $- x^{3} + 13 x^{2} - 55 x + 75$ by $x - 3$ .

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Step 4.1.4.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$3$

$x^{3}$

$13 x^{2}$

$55 x$

$75$

Step 4.1.4.5.2

Divide the highest order term in the dividend $- x^{3}$ by the highest order term in divisor $x$ .

				-	$x^{2}$
	$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$

Step 4.1.4.5.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			-	$x^{3}$	+	$3 x^{2}$

Step 4.1.4.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{3} + 3 x^{2}$

			-	$x^{2}$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			+	$x^{3}$	-	$3 x^{2}$

Step 4.1.4.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$x^{2}$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			+	$x^{3}$	-	$3 x^{2}$
					+	$10 x^{2}$

Step 4.1.4.5.6

Pull the next terms from the original dividend down into the current dividend.

			-	$x^{2}$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			+	$x^{3}$	-	$3 x^{2}$
					+	$10 x^{2}$	-	$55 x$

Step 4.1.4.5.7

Divide the highest order term in the dividend $10 x^{2}$ by the highest order term in divisor $x$ .

			-	$x^{2}$	+	$10 x$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			+	$x^{3}$	-	$3 x^{2}$
					+	$10 x^{2}$	-	$55 x$

Step 4.1.4.5.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$10 x$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			+	$x^{3}$	-	$3 x^{2}$
					+	$10 x^{2}$	-	$55 x$
					+	$10 x^{2}$	-	$30 x$

Step 4.1.4.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{2} - 30 x$

			-	$x^{2}$	+	$10 x$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			+	$x^{3}$	-	$3 x^{2}$
					+	$10 x^{2}$	-	$55 x$
					-	$10 x^{2}$	+	$30 x$

Step 4.1.4.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$x^{2}$	+	$10 x$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			+	$x^{3}$	-	$3 x^{2}$
					+	$10 x^{2}$	-	$55 x$
					-	$10 x^{2}$	+	$30 x$
							-	$25 x$

Step 4.1.4.5.11

Pull the next terms from the original dividend down into the current dividend.

			-	$x^{2}$	+	$10 x$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			+	$x^{3}$	-	$3 x^{2}$
					+	$10 x^{2}$	-	$55 x$
					-	$10 x^{2}$	+	$30 x$
							-	$25 x$	+	$75$

Step 4.1.4.5.12

Divide the highest order term in the dividend $- 25 x$ by the highest order term in divisor $x$ .

			-	$x^{2}$	+	$10 x$	-	$25$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			+	$x^{3}$	-	$3 x^{2}$
					+	$10 x^{2}$	-	$55 x$
					-	$10 x^{2}$	+	$30 x$
							-	$25 x$	+	$75$

Step 4.1.4.5.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$10 x$	-	$25$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			+	$x^{3}$	-	$3 x^{2}$
					+	$10 x^{2}$	-	$55 x$
					-	$10 x^{2}$	+	$30 x$
							-	$25 x$	+	$75$
							-	$25 x$	+	$75$

Step 4.1.4.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 25 x + 75$

			-	$x^{2}$	+	$10 x$	-	$25$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			+	$x^{3}$	-	$3 x^{2}$
					+	$10 x^{2}$	-	$55 x$
					-	$10 x^{2}$	+	$30 x$
							-	$25 x$	+	$75$
							+	$25 x$	-	$75$

Step 4.1.4.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$x^{2}$	+	$10 x$	-	$25$
$x$	-	$3$	-	$x^{3}$	+	$13 x^{2}$	-	$55 x$	+	$75$
			+	$x^{3}$	-	$3 x^{2}$
					+	$10 x^{2}$	-	$55 x$
					-	$10 x^{2}$	+	$30 x$
							-	$25 x$	+	$75$
							+	$25 x$	-	$75$
										$0$

Step 4.1.4.5.16

Since the remander is $0$ , the final answer is the quotient.

$- x^{2} + 10 x - 25$

Step 4.1.4.6

Write $- x^{3} + 13 x^{2} - 55 x + 75$ as a set of factors.

$6 ((x - 1) ((x - 1) ((x - 3) (- x^{2} + 10 x - 25)))) = 0$

Step 4.1.5

Factor by grouping.

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Step 4.1.5.1

Factor by grouping.

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Step 4.1.5.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 25 = 25$ and whose sum is $b = 10$ .

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Step 4.1.5.1.1.1

Factor $10$ out of $10 x$ .

$6 ((x - 1) ((x - 1) ((x - 3) (- x^{2} + 10 (x) - 25)))) = 0$

Step 4.1.5.1.1.2

Rewrite $10$ as $5$ plus $5$

$6 ((x - 1) ((x - 1) ((x - 3) (- x^{2} + (5 + 5) x - 25)))) = 0$

Step 4.1.5.1.1.3

Apply the distributive property.

$6 ((x - 1) ((x - 1) ((x - 3) (- x^{2} + 5 x + 5 x - 25)))) = 0$

Step 4.1.5.1.2

Factor out the greatest common factor from each group.

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Step 4.1.5.1.2.1

Group the first two terms and the last two terms.

$6 ((x - 1) ((x - 1) ((x - 3) ((- x^{2} + 5 x) + 5 x - 25)))) = 0$

Step 4.1.5.1.2.2

Factor out the greatest common factor (GCF) from each group.

$6 ((x - 1) ((x - 1) ((x - 3) (x (- x + 5) - 5 (- x + 5))))) = 0$

Step 4.1.5.1.3

Factor the polynomial by factoring out the greatest common factor, $- x + 5$ .

$6 ((x - 1) ((x - 1) ((x - 3) ((- x + 5) (x - 5))))) = 0$

Step 4.1.5.2

Remove unnecessary parentheses.

$6 ((x - 1) ((x - 1) ((x - 3) (- x + 5) (x - 5)))) = 0$

Step 4.1.6

Combine exponents.

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Step 4.1.6.1

Factor $- 1$ out of $- x$ .

$6 ((x - 1) ((x - 1) ((x - 3) (- (x) + 5) (x - 5)))) = 0$

Step 4.1.6.2

Rewrite $5$ as $- 1 (- 5)$ .

$6 ((x - 1) ((x - 1) ((x - 3) (- (x) - 1 \cdot - 5) (x - 5)))) = 0$

Step 4.1.6.3

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$6 ((x - 1) ((x - 1) ((x - 3) (- (x - 5)) (x - 5)))) = 0$

Step 4.1.6.4

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$6 ((x - 1) ((x - 1) ((x - 3) (- 1 (x - 5)) (x - 5)))) = 0$

Step 4.1.6.5

Remove parentheses.

$6 ((x - 1) ((x - 1) ((x - 3) \cdot (- 1 (x - 5) (x - 5))))) = 0$

Step 4.1.6.6

Raise $x - 5$ to the power of $1$ .

$6 ((x - 1) ((x - 1) ((x - 3) \cdot (- 1 ((x - 5) (x - 5)))))) = 0$

Step 4.1.6.7

Raise $x - 5$ to the power of $1$ .

$6 ((x - 1) ((x - 1) ((x - 3) \cdot (- 1 ((x - 5) (x - 5)))))) = 0$

Step 4.1.6.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$6 ((x - 1) ((x - 1) ((x - 3) \cdot (- 1 {(x - 5)}^{1 + 1})))) = 0$

Step 4.1.6.9

Add $1$ and $1$ .

$6 ((x - 1) ((x - 1) ((x - 3) \cdot (- 1 {(x - 5)}^{2})))) = 0$

Step 4.1.7

Factor.

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Step 4.1.7.1

Factor.

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Step 4.1.7.1.1

Factor out negative.

$6 ((x - 1) ((x - 1) (- ((x - 3) {(x - 5)}^{2})))) = 0$

Step 4.1.7.1.2

Remove unnecessary parentheses.

$6 ((x - 1) ((x - 1) (- (x - 3) {(x - 5)}^{2}))) = 0$

Step 4.1.7.2

Remove unnecessary parentheses.

$6 ((x - 1) ((x - 1) \cdot (- 1 (x - 3) {(x - 5)}^{2}))) = 0$

Step 4.1.8

Factor.

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Step 4.1.8.1

Factor.

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Step 4.1.8.1.1

Factor.

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Step 4.1.8.1.1.1

Factor out negative.

$6 ((x - 1) (- (((x - 1) (x - 3)) {(x - 5)}^{2}))) = 0$

Step 4.1.8.1.1.2

Remove unnecessary parentheses.

$6 ((x - 1) (- (x - 1) (x - 3) {(x - 5)}^{2})) = 0$

Step 4.1.8.1.2

Remove unnecessary parentheses.

$6 ((x - 1) \cdot (- 1 (x - 1) (x - 3) {(x - 5)}^{2})) = 0$

Step 4.1.8.2

Remove unnecessary parentheses.

$6 (x - 1) \cdot (- 1 (x - 1) (x - 3) {(x - 5)}^{2}) = 0$

Step 4.1.9

Combine exponents.

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Step 4.1.9.1

Multiply $- 1$ by $6$ .

$- 6 (x - 1) (x - 1) (x - 3) {(x - 5)}^{2} = 0$

Step 4.1.9.2

Raise $x - 1$ to the power of $1$ .

$- 6 ((x - 1) (x - 1)) (x - 3) {(x - 5)}^{2} = 0$

Step 4.1.9.3

Raise $x - 1$ to the power of $1$ .

$- 6 ((x - 1) (x - 1)) (x - 3) {(x - 5)}^{2} = 0$

Step 4.1.9.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 6 {(x - 1)}^{1 + 1} (x - 3) {(x - 5)}^{2} = 0$

Step 4.1.9.5

Add $1$ and $1$ .

$- 6 {(x - 1)}^{2} (x - 3) {(x - 5)}^{2} = 0$

Step 4.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x - 1)}^{2} = 0$

$x - 3 = 0$

${(x - 5)}^{2} = 0$

Step 4.3

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set ${(x - 1)}^{2}$ equal to $0$ .

${(x - 1)}^{2} = 0$

Step 4.3.2

Solve ${(x - 1)}^{2} = 0$ for $x$ .

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Step 4.3.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.3.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 4.4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.5

Set ${(x - 5)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set ${(x - 5)}^{2}$ equal to $0$ .

${(x - 5)}^{2} = 0$

Step 4.5.2

Solve ${(x - 5)}^{2} = 0$ for $x$ .

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Step 4.5.2.1

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.5.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.6

The final solution is all the values that make $- 6 {(x - 1)}^{2} (x - 3) {(x - 5)}^{2} = 0$ true.

$x = 1, 3, 5$

Step 5

Solve the original function $f (x) = - x^{6} + 18 x^{5} - 123 x^{4} + 396 x^{3} - 615 x^{2} + 450 x - 125$ at $x = 1$ .

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Step 5.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = - {(1)}^{6} + 18 {(1)}^{5} - 123 {(1)}^{4} + 396 {(1)}^{3} - 615 {(1)}^{2} + 450 (1) - 125$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

One to any power is one.

$f (1) = - 1 \cdot 1 + 18 {(1)}^{5} - 123 {(1)}^{4} + 396 {(1)}^{3} - 615 {(1)}^{2} + 450 (1) - 125$

Step 5.2.1.2

Multiply $- 1$ by $1$ .

$f (1) = - 1 + 18 {(1)}^{5} - 123 {(1)}^{4} + 396 {(1)}^{3} - 615 {(1)}^{2} + 450 (1) - 125$

Step 5.2.1.3

One to any power is one.

$f (1) = - 1 + 18 \cdot 1 - 123 {(1)}^{4} + 396 {(1)}^{3} - 615 {(1)}^{2} + 450 (1) - 125$

Step 5.2.1.4

Multiply $18$ by $1$ .

$f (1) = - 1 + 18 - 123 {(1)}^{4} + 396 {(1)}^{3} - 615 {(1)}^{2} + 450 (1) - 125$

Step 5.2.1.5

One to any power is one.

$f (1) = - 1 + 18 - 123 \cdot 1 + 396 {(1)}^{3} - 615 {(1)}^{2} + 450 (1) - 125$

Step 5.2.1.6

Multiply $- 123$ by $1$ .

$f (1) = - 1 + 18 - 123 + 396 {(1)}^{3} - 615 {(1)}^{2} + 450 (1) - 125$

Step 5.2.1.7

One to any power is one.

$f (1) = - 1 + 18 - 123 + 396 \cdot 1 - 615 {(1)}^{2} + 450 (1) - 125$

Step 5.2.1.8

Multiply $396$ by $1$ .

$f (1) = - 1 + 18 - 123 + 396 - 615 {(1)}^{2} + 450 (1) - 125$

Step 5.2.1.9

One to any power is one.

$f (1) = - 1 + 18 - 123 + 396 - 615 \cdot 1 + 450 (1) - 125$

Step 5.2.1.10

Multiply $- 615$ by $1$ .

$f (1) = - 1 + 18 - 123 + 396 - 615 + 450 (1) - 125$

Step 5.2.1.11

Multiply $450$ by $1$ .

$f (1) = - 1 + 18 - 123 + 396 - 615 + 450 - 125$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $- 1$ and $18$ .

$f (1) = 17 - 123 + 396 - 615 + 450 - 125$

Step 5.2.2.2

Subtract $123$ from $17$ .

$f (1) = - 106 + 396 - 615 + 450 - 125$

Step 5.2.2.3

Add $- 106$ and $396$ .

$f (1) = 290 - 615 + 450 - 125$

Step 5.2.2.4

Subtract $615$ from $290$ .

$f (1) = - 325 + 450 - 125$

Step 5.2.2.5

Add $- 325$ and $450$ .

$f (1) = 125 - 125$

Step 5.2.2.6

Subtract $125$ from $125$ .

$f (1) = 0$

Step 5.2.3

The final answer is $0$ .

$0$

Step 6

Solve the original function $f (x) = - x^{6} + 18 x^{5} - 123 x^{4} + 396 x^{3} - 615 x^{2} + 450 x - 125$ at $x = 3$ .

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Step 6.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = - {(3)}^{6} + 18 {(3)}^{5} - 123 {(3)}^{4} + 396 {(3)}^{3} - 615 {(3)}^{2} + 450 (3) - 125$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $3$ to the power of $6$ .

$f (3) = - 1 \cdot 729 + 18 {(3)}^{5} - 123 {(3)}^{4} + 396 {(3)}^{3} - 615 {(3)}^{2} + 450 (3) - 125$

Step 6.2.1.2

Multiply $- 1$ by $729$ .

$f (3) = - 729 + 18 {(3)}^{5} - 123 {(3)}^{4} + 396 {(3)}^{3} - 615 {(3)}^{2} + 450 (3) - 125$

Step 6.2.1.3

Raise $3$ to the power of $5$ .

$f (3) = - 729 + 18 \cdot 243 - 123 {(3)}^{4} + 396 {(3)}^{3} - 615 {(3)}^{2} + 450 (3) - 125$

Step 6.2.1.4

Multiply $18$ by $243$ .

$f (3) = - 729 + 4374 - 123 {(3)}^{4} + 396 {(3)}^{3} - 615 {(3)}^{2} + 450 (3) - 125$

Step 6.2.1.5

Raise $3$ to the power of $4$ .

$f (3) = - 729 + 4374 - 123 \cdot 81 + 396 {(3)}^{3} - 615 {(3)}^{2} + 450 (3) - 125$

Step 6.2.1.6

Multiply $- 123$ by $81$ .

$f (3) = - 729 + 4374 - 9963 + 396 {(3)}^{3} - 615 {(3)}^{2} + 450 (3) - 125$

Step 6.2.1.7

Raise $3$ to the power of $3$ .

$f (3) = - 729 + 4374 - 9963 + 396 \cdot 27 - 615 {(3)}^{2} + 450 (3) - 125$

Step 6.2.1.8

Multiply $396$ by $27$ .

$f (3) = - 729 + 4374 - 9963 + 10692 - 615 {(3)}^{2} + 450 (3) - 125$

Step 6.2.1.9

Raise $3$ to the power of $2$ .

$f (3) = - 729 + 4374 - 9963 + 10692 - 615 \cdot 9 + 450 (3) - 125$

Step 6.2.1.10

Multiply $- 615$ by $9$ .

$f (3) = - 729 + 4374 - 9963 + 10692 - 5535 + 450 (3) - 125$

Step 6.2.1.11

Multiply $450$ by $3$ .

$f (3) = - 729 + 4374 - 9963 + 10692 - 5535 + 1350 - 125$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $- 729$ and $4374$ .

$f (3) = 3645 - 9963 + 10692 - 5535 + 1350 - 125$

Step 6.2.2.2

Subtract $9963$ from $3645$ .

$f (3) = - 6318 + 10692 - 5535 + 1350 - 125$

Step 6.2.2.3

Add $- 6318$ and $10692$ .

$f (3) = 4374 - 5535 + 1350 - 125$

Step 6.2.2.4

Subtract $5535$ from $4374$ .

$f (3) = - 1161 + 1350 - 125$

Step 6.2.2.5

Add $- 1161$ and $1350$ .

$f (3) = 189 - 125$

Step 6.2.2.6

Subtract $125$ from $189$ .

$f (3) = 64$

Step 6.2.3

The final answer is $64$ .

$64$

Step 7

Solve the original function $f (x) = - x^{6} + 18 x^{5} - 123 x^{4} + 396 x^{3} - 615 x^{2} + 450 x - 125$ at $x = 5$ .

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Step 7.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = - {(5)}^{6} + 18 {(5)}^{5} - 123 {(5)}^{4} + 396 {(5)}^{3} - 615 {(5)}^{2} + 450 (5) - 125$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $5$ to the power of $6$ .

$f (5) = - 1 \cdot 15625 + 18 {(5)}^{5} - 123 {(5)}^{4} + 396 {(5)}^{3} - 615 {(5)}^{2} + 450 (5) - 125$

Step 7.2.1.2

Multiply $- 1$ by $15625$ .

$f (5) = - 15625 + 18 {(5)}^{5} - 123 {(5)}^{4} + 396 {(5)}^{3} - 615 {(5)}^{2} + 450 (5) - 125$

Step 7.2.1.3

Raise $5$ to the power of $5$ .

$f (5) = - 15625 + 18 \cdot 3125 - 123 {(5)}^{4} + 396 {(5)}^{3} - 615 {(5)}^{2} + 450 (5) - 125$

Step 7.2.1.4

Multiply $18$ by $3125$ .

$f (5) = - 15625 + 56250 - 123 {(5)}^{4} + 396 {(5)}^{3} - 615 {(5)}^{2} + 450 (5) - 125$

Step 7.2.1.5

Raise $5$ to the power of $4$ .

$f (5) = - 15625 + 56250 - 123 \cdot 625 + 396 {(5)}^{3} - 615 {(5)}^{2} + 450 (5) - 125$

Step 7.2.1.6

Multiply $- 123$ by $625$ .

$f (5) = - 15625 + 56250 - 76875 + 396 {(5)}^{3} - 615 {(5)}^{2} + 450 (5) - 125$

Step 7.2.1.7

Raise $5$ to the power of $3$ .

$f (5) = - 15625 + 56250 - 76875 + 396 \cdot 125 - 615 {(5)}^{2} + 450 (5) - 125$

Step 7.2.1.8

Multiply $396$ by $125$ .

$f (5) = - 15625 + 56250 - 76875 + 49500 - 615 {(5)}^{2} + 450 (5) - 125$

Step 7.2.1.9

Raise $5$ to the power of $2$ .

$f (5) = - 15625 + 56250 - 76875 + 49500 - 615 \cdot 25 + 450 (5) - 125$

Step 7.2.1.10

Multiply $- 615$ by $25$ .

$f (5) = - 15625 + 56250 - 76875 + 49500 - 15375 + 450 (5) - 125$

Step 7.2.1.11

Multiply $450$ by $5$ .

$f (5) = - 15625 + 56250 - 76875 + 49500 - 15375 + 2250 - 125$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Add $- 15625$ and $56250$ .

$f (5) = 40625 - 76875 + 49500 - 15375 + 2250 - 125$

Step 7.2.2.2

Subtract $76875$ from $40625$ .

$f (5) = - 36250 + 49500 - 15375 + 2250 - 125$

Step 7.2.2.3

Add $- 36250$ and $49500$ .

$f (5) = 13250 - 15375 + 2250 - 125$

Step 7.2.2.4

Subtract $15375$ from $13250$ .

$f (5) = - 2125 + 2250 - 125$

Step 7.2.2.5

Add $- 2125$ and $2250$ .

$f (5) = 125 - 125$

Step 7.2.2.6

Subtract $125$ from $125$ .

$f (5) = 0$

Step 7.2.3

The final answer is $0$ .

$0$

Step 8

The horizontal tangent lines on function $f (x) = - x^{6} + 18 x^{5} - 123 x^{4} + 396 x^{3} - 615 x^{2} + 450 x - 125$ are $y = 0, y = 64, y = 0$ .

$y = 0, y = 64, y = 0$

Step 9