Precalculus Examples

${(p - q)}^{6}$

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 ${(p - q)}^{6}$ , $n = 6$ 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 $a$ $p$ and $b$ $- q$ into the expression.

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

Apply the product rule to $- q$ .

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

Step 4.3

Rewrite using the commutative property of multiplication.

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

Step 4.4

Anything raised to $0$ is $1$ .

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

Step 4.5

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

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

Step 4.6

Anything raised to $0$ is $1$ .

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

Step 4.7

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

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

Step 4.8

Simplify.

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

Step 4.9

Rewrite using the commutative property of multiplication.

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

Step 4.10

Multiply $6$ by $- 1$ .

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

Step 4.11

Apply the product rule to $- q$ .

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

Step 4.12

Rewrite using the commutative property of multiplication.

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

Step 4.13

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

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

Step 4.14

Multiply $15$ by $1$ .

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

Step 4.15

Apply the product rule to $- q$ .

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

Step 4.16

Rewrite using the commutative property of multiplication.

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

Step 4.17

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

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

Step 4.18

Multiply $20$ by $- 1$ .

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

Step 4.19

Apply the product rule to $- q$ .

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

Step 4.20

Rewrite using the commutative property of multiplication.

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

Step 4.21

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

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

Step 4.22

Multiply $15$ by $1$ .

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

Step 4.23

Simplify.

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

Step 4.24

Apply the product rule to $- q$ .

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

Step 4.25

Rewrite using the commutative property of multiplication.

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

Step 4.26

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

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

Step 4.27

Multiply $6$ by $- 1$ .

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

Step 4.28

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

$p^{6} - 6 p^{5} q + 15 p^{4} q^{2} - 20 p^{3} q^{3} + 15 p^{2} q^{4} - 6 p q^{5} + {(p)}^{0} {(- q)}^{6}$

Step 4.29

Anything raised to $0$ is $1$ .

$p^{6} - 6 p^{5} q + 15 p^{4} q^{2} - 20 p^{3} q^{3} + 15 p^{2} q^{4} - 6 p q^{5} + 1 {(- q)}^{6}$

Step 4.30

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

$p^{6} - 6 p^{5} q + 15 p^{4} q^{2} - 20 p^{3} q^{3} + 15 p^{2} q^{4} - 6 p q^{5} + {(- q)}^{6}$

Step 4.31

Apply the product rule to $- q$ .

$p^{6} - 6 p^{5} q + 15 p^{4} q^{2} - 20 p^{3} q^{3} + 15 p^{2} q^{4} - 6 p q^{5} + {(- 1)}^{6} q^{6}$

Step 4.32

Raise $- 1$ to the power of $6$ .

$p^{6} - 6 p^{5} q + 15 p^{4} q^{2} - 20 p^{3} q^{3} + 15 p^{2} q^{4} - 6 p q^{5} + 1 q^{6}$

Step 4.33

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

$p^{6} - 6 p^{5} q + 15 p^{4} q^{2} - 20 p^{3} q^{3} + 15 p^{2} q^{4} - 6 p q^{5} + q^{6}$