Algebra Examples

$f (x) = {(x + 4)}^{2} (x + 2) {(x - 2)}^{2}$

Step 1

Identify the degree of the function.

Tap for more steps...

Step 1.1

Simplify and reorder the polynomial.

Tap for more steps...

Step 1.1.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$(x + 4) (x + 4) (x + 2) {(x - 2)}^{2}$

Step 1.1.2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

Tap for more steps...

Step 1.1.2.1

Apply the distributive property.

$(x (x + 4) + 4 (x + 4)) (x + 2) {(x - 2)}^{2}$

Step 1.1.2.2

Apply the distributive property.

$(x \cdot x + x \cdot 4 + 4 (x + 4)) (x + 2) {(x - 2)}^{2}$

Step 1.1.2.3

Apply the distributive property.

$(x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) (x + 2) {(x - 2)}^{2}$

Step 1.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.1.3.1

Simplify each term.

Tap for more steps...

Step 1.1.3.1.1

Multiply $x$ by $x$ .

$(x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) (x + 2) {(x - 2)}^{2}$

Step 1.1.3.1.2

Move $4$ to the left of $x$ .

$(x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) (x + 2) {(x - 2)}^{2}$

Step 1.1.3.1.3

Multiply $4$ by $4$ .

$(x^{2} + 4 x + 4 x + 16) (x + 2) {(x - 2)}^{2}$

Step 1.1.3.2

Add $4 x$ and $4 x$ .

$(x^{2} + 8 x + 16) (x + 2) {(x - 2)}^{2}$

Step 1.1.4

Expand $(x^{2} + 8 x + 16) (x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$(x^{2} x + x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.1.5

Simplify terms.

Tap for more steps...

Step 1.1.5.1

Simplify each term.

Tap for more steps...

Step 1.1.5.1.1

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

Tap for more steps...

Step 1.1.5.1.1.1

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

Tap for more steps...

Step 1.1.5.1.1.1.1

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

$(x^{2} x^{1} + x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.1.5.1.1.1.2

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

$(x^{2 + 1} + x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.1.5.1.1.2

Add $2$ and $1$ .

$(x^{3} + x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.1.5.1.2

Move $2$ to the left of $x^{2}$ .

$(x^{3} + 2 \cdot x^{2} + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.1.5.1.3

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.5.1.3.1

Move $x$ .

$(x^{3} + 2 x^{2} + 8 (x \cdot x) + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.1.5.1.3.2

Multiply $x$ by $x$ .

$(x^{3} + 2 x^{2} + 8 x^{2} + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.1.5.1.4

Multiply $2$ by $8$ .

$(x^{3} + 2 x^{2} + 8 x^{2} + 16 x + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.1.5.1.5

Multiply $16$ by $2$ .

$(x^{3} + 2 x^{2} + 8 x^{2} + 16 x + 16 x + 32) {(x - 2)}^{2}$

Step 1.1.5.2

Simplify by adding terms.

Tap for more steps...

Step 1.1.5.2.1

Add $2 x^{2}$ and $8 x^{2}$ .

$(x^{3} + 10 x^{2} + 16 x + 16 x + 32) {(x - 2)}^{2}$

Step 1.1.5.2.2

Add $16 x$ and $16 x$ .

$(x^{3} + 10 x^{2} + 32 x + 32) {(x - 2)}^{2}$

Step 1.1.5.2.3

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$(x^{3} + 10 x^{2} + 32 x + 32) ((x - 2) (x - 2))$

Step 1.1.6

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Tap for more steps...

Step 1.1.6.1

Apply the distributive property.

$(x^{3} + 10 x^{2} + 32 x + 32) (x (x - 2) - 2 (x - 2))$

Step 1.1.6.2

Apply the distributive property.

$(x^{3} + 10 x^{2} + 32 x + 32) (x \cdot x + x \cdot - 2 - 2 (x - 2))$

Step 1.1.6.3

Apply the distributive property.

$(x^{3} + 10 x^{2} + 32 x + 32) (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2)$

Step 1.1.7

Simplify and combine like terms.

Tap for more steps...

Step 1.1.7.1

Simplify each term.

Tap for more steps...

Step 1.1.7.1.1

Multiply $x$ by $x$ .

$(x^{3} + 10 x^{2} + 32 x + 32) (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2)$

Step 1.1.7.1.2

Move $- 2$ to the left of $x$ .

$(x^{3} + 10 x^{2} + 32 x + 32) (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2)$

Step 1.1.7.1.3

Multiply $- 2$ by $- 2$ .

$(x^{3} + 10 x^{2} + 32 x + 32) (x^{2} - 2 x - 2 x + 4)$

Step 1.1.7.2

Subtract $2 x$ from $- 2 x$ .

$(x^{3} + 10 x^{2} + 32 x + 32) (x^{2} - 4 x + 4)$

Step 1.1.8

Expand $(x^{3} + 10 x^{2} + 32 x + 32) (x^{2} - 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x \cdot 4 + 32 x^{2} + 32 (- 4 x) + 32 \cdot 4$

Step 1.1.9

Simplify terms.

Tap for more steps...

Step 1.1.9.1

Combine the opposite terms in $x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x \cdot 4 + 32 x^{2} + 32 (- 4 x) + 32 \cdot 4$ .

Tap for more steps...

Step 1.1.9.1.1

Reorder the factors in the terms $32 x \cdot 4$ and $32 (- 4 x)$ .

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 4 \cdot 32 x + 32 x^{2} - 4 \cdot 32 x + 32 \cdot 4$

Step 1.1.9.1.2

Subtract $4 \cdot 32 x$ from $4 \cdot 32 x$ .

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 0 + 32 \cdot 4$

Step 1.1.9.1.3

Add $x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2}$ and $0$ .

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2

Simplify each term.

Tap for more steps...

Step 1.1.9.2.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.1.9.2.1.1

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

$x^{3 + 2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.1.2

Add $3$ and $2$ .

$x^{5} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.2

Rewrite using the commutative property of multiplication.

$x^{5} - 4 x^{3} x + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.3

Multiply $x^{3}$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.9.2.3.1

Move $x$ .

$x^{5} - 4 (x \cdot x^{3}) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.3.2

Multiply $x$ by $x^{3}$ .

Tap for more steps...

Step 1.1.9.2.3.2.1

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

$x^{5} - 4 (x^{1} x^{3}) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.3.2.2

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

$x^{5} - 4 x^{1 + 3} + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.3.3

Add $1$ and $3$ .

$x^{5} - 4 x^{4} + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.4

Move $4$ to the left of $x^{3}$ .

$x^{5} - 4 x^{4} + 4 \cdot x^{3} + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.5

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

Tap for more steps...

Step 1.1.9.2.5.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 (x^{2} x^{2}) + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.5.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{2 + 2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.5.3

Add $2$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.6

Rewrite using the commutative property of multiplication.

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot - 4 x^{2} x + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.7

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

Tap for more steps...

Step 1.1.9.2.7.1

Move $x$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot - 4 (x \cdot x^{2}) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.7.2

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

Tap for more steps...

Step 1.1.9.2.7.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot - 4 (x^{1} x^{2}) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.7.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot - 4 x^{1 + 2} + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.7.3

Add $1$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot - 4 x^{3} + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.8

Multiply $10$ by $- 4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.9

Multiply $4$ by $10$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.10

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

Tap for more steps...

Step 1.1.9.2.10.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 (x^{2} x) + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.10.2

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

Tap for more steps...

Step 1.1.9.2.10.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 (x^{2} x^{1}) + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.10.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{2 + 1} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.10.3

Add $2$ and $1$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.11

Rewrite using the commutative property of multiplication.

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} + 32 \cdot - 4 x \cdot x + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.12

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.9.2.12.1

Move $x$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} + 32 \cdot - 4 (x \cdot x) + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.12.2

Multiply $x$ by $x$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} + 32 \cdot - 4 x^{2} + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.13

Multiply $32$ by $- 4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} - 128 x^{2} + 32 x^{2} + 32 \cdot 4$

Step 1.1.9.2.14

Multiply $32$ by $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} - 128 x^{2} + 32 x^{2} + 128$

Step 1.1.9.3

Simplify by adding terms.

Tap for more steps...

Step 1.1.9.3.1

Add $- 4 x^{4}$ and $10 x^{4}$ .

$x^{5} + 6 x^{4} + 4 x^{3} - 40 x^{3} + 40 x^{2} + 32 x^{3} - 128 x^{2} + 32 x^{2} + 128$

Step 1.1.9.3.2

Subtract $40 x^{3}$ from $4 x^{3}$ .

$x^{5} + 6 x^{4} - 36 x^{3} + 40 x^{2} + 32 x^{3} - 128 x^{2} + 32 x^{2} + 128$

Step 1.1.9.3.3

Add $- 36 x^{3}$ and $32 x^{3}$ .

$x^{5} + 6 x^{4} - 4 x^{3} + 40 x^{2} - 128 x^{2} + 32 x^{2} + 128$

Step 1.1.9.3.4

Subtract $128 x^{2}$ from $40 x^{2}$ .

$x^{5} + 6 x^{4} - 4 x^{3} - 88 x^{2} + 32 x^{2} + 128$

Step 1.1.9.3.5

Add $- 88 x^{2}$ and $32 x^{2}$ .

$x^{5} + 6 x^{4} - 4 x^{3} - 56 x^{2} + 128$

Step 1.2

The largest exponent is the degree of the polynomial.

$5$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

Tap for more steps...

Step 3.1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

Tap for more steps...

Step 3.1.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$(x + 4) (x + 4) (x + 2) {(x - 2)}^{2}$

Step 3.1.2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

Tap for more steps...

Step 3.1.2.1

Apply the distributive property.

$(x (x + 4) + 4 (x + 4)) (x + 2) {(x - 2)}^{2}$

Step 3.1.2.2

Apply the distributive property.

$(x \cdot x + x \cdot 4 + 4 (x + 4)) (x + 2) {(x - 2)}^{2}$

Step 3.1.2.3

Apply the distributive property.

$(x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) (x + 2) {(x - 2)}^{2}$

Step 3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.1.3.1.1

Multiply $x$ by $x$ .

$(x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) (x + 2) {(x - 2)}^{2}$

Step 3.1.3.1.2

Move $4$ to the left of $x$ .

$(x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) (x + 2) {(x - 2)}^{2}$

Step 3.1.3.1.3

Multiply $4$ by $4$ .

$(x^{2} + 4 x + 4 x + 16) (x + 2) {(x - 2)}^{2}$

Step 3.1.3.2

Add $4 x$ and $4 x$ .

$(x^{2} + 8 x + 16) (x + 2) {(x - 2)}^{2}$

Step 3.1.4

Expand $(x^{2} + 8 x + 16) (x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$(x^{2} x + x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 3.1.5

Simplify terms.

Tap for more steps...

Step 3.1.5.1

Simplify each term.

Tap for more steps...

Step 3.1.5.1.1

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

Tap for more steps...

Step 3.1.5.1.1.1

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

Tap for more steps...

Step 3.1.5.1.1.1.1

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

$(x^{2} x^{1} + x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 3.1.5.1.1.1.2

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

$(x^{2 + 1} + x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 3.1.5.1.1.2

Add $2$ and $1$ .

$(x^{3} + x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 3.1.5.1.2

Move $2$ to the left of $x^{2}$ .

$(x^{3} + 2 \cdot x^{2} + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 3.1.5.1.3

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.1.5.1.3.1

Move $x$ .

$(x^{3} + 2 x^{2} + 8 (x \cdot x) + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 3.1.5.1.3.2

Multiply $x$ by $x$ .

$(x^{3} + 2 x^{2} + 8 x^{2} + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 3.1.5.1.4

Multiply $2$ by $8$ .

$(x^{3} + 2 x^{2} + 8 x^{2} + 16 x + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 3.1.5.1.5

Multiply $16$ by $2$ .

$(x^{3} + 2 x^{2} + 8 x^{2} + 16 x + 16 x + 32) {(x - 2)}^{2}$

Step 3.1.5.2

Simplify by adding terms.

Tap for more steps...

Step 3.1.5.2.1

Add $2 x^{2}$ and $8 x^{2}$ .

$(x^{3} + 10 x^{2} + 16 x + 16 x + 32) {(x - 2)}^{2}$

Step 3.1.5.2.2

Add $16 x$ and $16 x$ .

$(x^{3} + 10 x^{2} + 32 x + 32) {(x - 2)}^{2}$

Step 3.1.5.2.3

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$(x^{3} + 10 x^{2} + 32 x + 32) ((x - 2) (x - 2))$

Step 3.1.6

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Tap for more steps...

Step 3.1.6.1

Apply the distributive property.

$(x^{3} + 10 x^{2} + 32 x + 32) (x (x - 2) - 2 (x - 2))$

Step 3.1.6.2

Apply the distributive property.

$(x^{3} + 10 x^{2} + 32 x + 32) (x \cdot x + x \cdot - 2 - 2 (x - 2))$

Step 3.1.6.3

Apply the distributive property.

$(x^{3} + 10 x^{2} + 32 x + 32) (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2)$

Step 3.1.7

Simplify and combine like terms.

Tap for more steps...

Step 3.1.7.1

Simplify each term.

Tap for more steps...

Step 3.1.7.1.1

Multiply $x$ by $x$ .

$(x^{3} + 10 x^{2} + 32 x + 32) (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2)$

Step 3.1.7.1.2

Move $- 2$ to the left of $x$ .

$(x^{3} + 10 x^{2} + 32 x + 32) (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2)$

Step 3.1.7.1.3

Multiply $- 2$ by $- 2$ .

$(x^{3} + 10 x^{2} + 32 x + 32) (x^{2} - 2 x - 2 x + 4)$

Step 3.1.7.2

Subtract $2 x$ from $- 2 x$ .

$(x^{3} + 10 x^{2} + 32 x + 32) (x^{2} - 4 x + 4)$

Step 3.1.8

Expand $(x^{3} + 10 x^{2} + 32 x + 32) (x^{2} - 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x \cdot 4 + 32 x^{2} + 32 (- 4 x) + 32 \cdot 4$

Step 3.1.9

Simplify terms.

Tap for more steps...

Step 3.1.9.1

Tap for more steps...

Step 3.1.9.1.1

Reorder the factors in the terms $32 x \cdot 4$ and $32 (- 4 x)$ .

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 4 \cdot 32 x + 32 x^{2} - 4 \cdot 32 x + 32 \cdot 4$

Step 3.1.9.1.2

Subtract $4 \cdot 32 x$ from $4 \cdot 32 x$ .

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 0 + 32 \cdot 4$

Step 3.1.9.1.3

Add $x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2}$ and $0$ .

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2

Simplify each term.

Tap for more steps...

Step 3.1.9.2.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.1.9.2.1.1

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

$x^{3 + 2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.1.2

Add $3$ and $2$ .

$x^{5} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.2

Rewrite using the commutative property of multiplication.

$x^{5} - 4 x^{3} x + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.3

Multiply $x^{3}$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.1.9.2.3.1

Move $x$ .

$x^{5} - 4 (x \cdot x^{3}) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.3.2

Multiply $x$ by $x^{3}$ .

Tap for more steps...

Step 3.1.9.2.3.2.1

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

$x^{5} - 4 (x^{1} x^{3}) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.3.2.2

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

$x^{5} - 4 x^{1 + 3} + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.3.3

Add $1$ and $3$ .

$x^{5} - 4 x^{4} + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.4

Move $4$ to the left of $x^{3}$ .

$x^{5} - 4 x^{4} + 4 \cdot x^{3} + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.5

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

Tap for more steps...

Step 3.1.9.2.5.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 (x^{2} x^{2}) + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.5.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{2 + 2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.5.3

Add $2$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.6

Rewrite using the commutative property of multiplication.

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot - 4 x^{2} x + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.7

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

Tap for more steps...

Step 3.1.9.2.7.1

Move $x$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot - 4 (x \cdot x^{2}) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.7.2

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

Tap for more steps...

Step 3.1.9.2.7.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot - 4 (x^{1} x^{2}) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.7.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot - 4 x^{1 + 2} + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.7.3

Add $1$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot - 4 x^{3} + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.8

Multiply $10$ by $- 4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.9

Multiply $4$ by $10$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.10

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

Tap for more steps...

Step 3.1.9.2.10.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 (x^{2} x) + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.10.2

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

Tap for more steps...

Step 3.1.9.2.10.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 (x^{2} x^{1}) + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.10.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{2 + 1} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.10.3

Add $2$ and $1$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.11

Rewrite using the commutative property of multiplication.

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} + 32 \cdot - 4 x \cdot x + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.12

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.1.9.2.12.1

Move $x$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} + 32 \cdot - 4 (x \cdot x) + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.12.2

Multiply $x$ by $x$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} + 32 \cdot - 4 x^{2} + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.13

Multiply $32$ by $- 4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} - 128 x^{2} + 32 x^{2} + 32 \cdot 4$

Step 3.1.9.2.14

Multiply $32$ by $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} - 128 x^{2} + 32 x^{2} + 128$

Step 3.1.9.3

Simplify by adding terms.

Tap for more steps...

Step 3.1.9.3.1

Add $- 4 x^{4}$ and $10 x^{4}$ .

$x^{5} + 6 x^{4} + 4 x^{3} - 40 x^{3} + 40 x^{2} + 32 x^{3} - 128 x^{2} + 32 x^{2} + 128$

Step 3.1.9.3.2

Subtract $40 x^{3}$ from $4 x^{3}$ .

$x^{5} + 6 x^{4} - 36 x^{3} + 40 x^{2} + 32 x^{3} - 128 x^{2} + 32 x^{2} + 128$

Step 3.1.9.3.3

Add $- 36 x^{3}$ and $32 x^{3}$ .

$x^{5} + 6 x^{4} - 4 x^{3} + 40 x^{2} - 128 x^{2} + 32 x^{2} + 128$

Step 3.1.9.3.4

Subtract $128 x^{2}$ from $40 x^{2}$ .

$x^{5} + 6 x^{4} - 4 x^{3} - 88 x^{2} + 32 x^{2} + 128$

Step 3.1.9.3.5

Add $- 88 x^{2}$ and $32 x^{2}$ .

$x^{5} + 6 x^{4} - 4 x^{3} - 56 x^{2} + 128$

Step 3.2

The leading term in a polynomial is the term with the highest degree.

$x^{5}$

Step 3.3

The leading coefficient in a polynomial is the coefficient of the leading term.

$1$

Step 4

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Falls to the left and rises to the right

Step 7