Algebra Examples

${(d - 5 y)}^{6}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

$1 - 5 - 10 - 10 - 5 - 1$

$1 - 6 - 15 - 20 - 15 - 6 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(d - 5 y)}^{6}$ , $n = 6$ so the coefficients of the expansion will correspond with line $7$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 6 - 15 - 20 - 15 - 6 - 1$ .

$1 a^{6} b^{0} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + 1 a^{0} b^{6}$

Step 3

Substitute the actual values of $a$ $d$ and $b$ $- 5 y$ into the expression.

$1 {(d)}^{6} {(- 5 y)}^{0} + 6 {(d)}^{5} {(- 5 y)}^{1} + 15 {(d)}^{4} {(- 5 y)}^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(d)}^{6}$ by $1$ .

${(d)}^{6} {(- 5 y)}^{0} + 6 {(d)}^{5} {(- 5 y)}^{1} + 15 {(d)}^{4} {(- 5 y)}^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.2

Apply the product rule to $- 5 y$ .

$d^{6} ({(- 5)}^{0} y^{0}) + 6 {(d)}^{5} {(- 5 y)}^{1} + 15 {(d)}^{4} {(- 5 y)}^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.3

Rewrite using the commutative property of multiplication.

${(- 5)}^{0} d^{6} y^{0} + 6 {(d)}^{5} {(- 5 y)}^{1} + 15 {(d)}^{4} {(- 5 y)}^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.4

Anything raised to $0$ is $1$ .

$1 d^{6} y^{0} + 6 {(d)}^{5} {(- 5 y)}^{1} + 15 {(d)}^{4} {(- 5 y)}^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.5

Multiply $d^{6}$ by $1$ .

$d^{6} y^{0} + 6 {(d)}^{5} {(- 5 y)}^{1} + 15 {(d)}^{4} {(- 5 y)}^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.6

Anything raised to $0$ is $1$ .

$d^{6} \cdot 1 + 6 {(d)}^{5} {(- 5 y)}^{1} + 15 {(d)}^{4} {(- 5 y)}^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.7

Multiply $d^{6}$ by $1$ .

$d^{6} + 6 {(d)}^{5} {(- 5 y)}^{1} + 15 {(d)}^{4} {(- 5 y)}^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.8

Simplify.

$d^{6} + 6 d^{5} (- 5 y) + 15 {(d)}^{4} {(- 5 y)}^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.9

Rewrite using the commutative property of multiplication.

$d^{6} + 6 \cdot - 5 d^{5} y + 15 {(d)}^{4} {(- 5 y)}^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.10

Multiply $6$ by $- 5$ .

$d^{6} - 30 d^{5} y + 15 {(d)}^{4} {(- 5 y)}^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.11

Apply the product rule to $- 5 y$ .

$d^{6} - 30 d^{5} y + 15 d^{4} ({(- 5)}^{2} y^{2}) + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.12

Rewrite using the commutative property of multiplication.

$d^{6} - 30 d^{5} y + 15 \cdot {(- 5)}^{2} d^{4} y^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.13

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

$d^{6} - 30 d^{5} y + 15 \cdot 25 d^{4} y^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.14

Multiply $15$ by $25$ .

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} + 20 {(d)}^{3} {(- 5 y)}^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.15

Apply the product rule to $- 5 y$ .

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} + 20 d^{3} ({(- 5)}^{3} y^{3}) + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.16

Rewrite using the commutative property of multiplication.

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} + 20 \cdot {(- 5)}^{3} d^{3} y^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.17

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

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} + 20 \cdot - 125 d^{3} y^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.18

Multiply $20$ by $- 125$ .

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 15 {(d)}^{2} {(- 5 y)}^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.19

Apply the product rule to $- 5 y$ .

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 15 d^{2} ({(- 5)}^{4} y^{4}) + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.20

Rewrite using the commutative property of multiplication.

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 15 \cdot {(- 5)}^{4} d^{2} y^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.21

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

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 15 \cdot 625 d^{2} y^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.22

Multiply $15$ by $625$ .

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 9375 d^{2} y^{4} + 6 {(d)}^{1} {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.23

Simplify.

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 9375 d^{2} y^{4} + 6 d {(- 5 y)}^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.24

Apply the product rule to $- 5 y$ .

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 9375 d^{2} y^{4} + 6 d ({(- 5)}^{5} y^{5}) + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.25

Rewrite using the commutative property of multiplication.

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 9375 d^{2} y^{4} + 6 \cdot {(- 5)}^{5} d y^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.26

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

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 9375 d^{2} y^{4} + 6 \cdot - 3125 d y^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.27

Multiply $6$ by $- 3125$ .

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 9375 d^{2} y^{4} - 18750 d y^{5} + 1 {(d)}^{0} {(- 5 y)}^{6}$

Step 4.28

Multiply ${(d)}^{0}$ by $1$ .

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 9375 d^{2} y^{4} - 18750 d y^{5} + {(d)}^{0} {(- 5 y)}^{6}$

Step 4.29

Anything raised to $0$ is $1$ .

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 9375 d^{2} y^{4} - 18750 d y^{5} + 1 {(- 5 y)}^{6}$

Step 4.30

Multiply ${(- 5 y)}^{6}$ by $1$ .

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 9375 d^{2} y^{4} - 18750 d y^{5} + {(- 5 y)}^{6}$

Step 4.31

Apply the product rule to $- 5 y$ .

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 9375 d^{2} y^{4} - 18750 d y^{5} + {(- 5)}^{6} y^{6}$

Step 4.32

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

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 9375 d^{2} y^{4} - 18750 d y^{5} + 15625 y^{6}$