Precalculus Examples

${(s - 5 v)}^{5}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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

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 ${(s - 5 v)}^{5}$ , $n = 5$ so the coefficients of the expansion will correspond with line $6$ .

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

$1 a^{5} b^{0} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + 1 a^{0} b^{5}$

Step 3

Substitute the actual values of $a$ $s$ and $b$ $- 5 v$ into the expression.

$1 {(s)}^{5} {(- 5 v)}^{0} + 5 {(s)}^{4} {(- 5 v)}^{1} + 10 {(s)}^{3} {(- 5 v)}^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(s)}^{5}$ by $1$ .

${(s)}^{5} {(- 5 v)}^{0} + 5 {(s)}^{4} {(- 5 v)}^{1} + 10 {(s)}^{3} {(- 5 v)}^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.2

Apply the product rule to $- 5 v$ .

$s^{5} ({(- 5)}^{0} v^{0}) + 5 {(s)}^{4} {(- 5 v)}^{1} + 10 {(s)}^{3} {(- 5 v)}^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.3

Rewrite using the commutative property of multiplication.

${(- 5)}^{0} s^{5} v^{0} + 5 {(s)}^{4} {(- 5 v)}^{1} + 10 {(s)}^{3} {(- 5 v)}^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.4

Anything raised to $0$ is $1$ .

$1 s^{5} v^{0} + 5 {(s)}^{4} {(- 5 v)}^{1} + 10 {(s)}^{3} {(- 5 v)}^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.5

Multiply $s^{5}$ by $1$ .

$s^{5} v^{0} + 5 {(s)}^{4} {(- 5 v)}^{1} + 10 {(s)}^{3} {(- 5 v)}^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.6

Anything raised to $0$ is $1$ .

$s^{5} \cdot 1 + 5 {(s)}^{4} {(- 5 v)}^{1} + 10 {(s)}^{3} {(- 5 v)}^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.7

Multiply $s^{5}$ by $1$ .

$s^{5} + 5 {(s)}^{4} {(- 5 v)}^{1} + 10 {(s)}^{3} {(- 5 v)}^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.8

Simplify.

$s^{5} + 5 s^{4} (- 5 v) + 10 {(s)}^{3} {(- 5 v)}^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.9

Rewrite using the commutative property of multiplication.

$s^{5} + 5 \cdot - 5 s^{4} v + 10 {(s)}^{3} {(- 5 v)}^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.10

Multiply $5$ by $- 5$ .

$s^{5} - 25 s^{4} v + 10 {(s)}^{3} {(- 5 v)}^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.11

Apply the product rule to $- 5 v$ .

$s^{5} - 25 s^{4} v + 10 s^{3} ({(- 5)}^{2} v^{2}) + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.12

Rewrite using the commutative property of multiplication.

$s^{5} - 25 s^{4} v + 10 \cdot {(- 5)}^{2} s^{3} v^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.13

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

$s^{5} - 25 s^{4} v + 10 \cdot 25 s^{3} v^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.14

Multiply $10$ by $25$ .

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} + 10 {(s)}^{2} {(- 5 v)}^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.15

Apply the product rule to $- 5 v$ .

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} + 10 s^{2} ({(- 5)}^{3} v^{3}) + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.16

Rewrite using the commutative property of multiplication.

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} + 10 \cdot {(- 5)}^{3} s^{2} v^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.17

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

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} + 10 \cdot - 125 s^{2} v^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.18

Multiply $10$ by $- 125$ .

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} - 1250 s^{2} v^{3} + 5 {(s)}^{1} {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.19

Simplify.

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} - 1250 s^{2} v^{3} + 5 s {(- 5 v)}^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.20

Apply the product rule to $- 5 v$ .

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} - 1250 s^{2} v^{3} + 5 s ({(- 5)}^{4} v^{4}) + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.21

Rewrite using the commutative property of multiplication.

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} - 1250 s^{2} v^{3} + 5 \cdot {(- 5)}^{4} s v^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.22

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

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} - 1250 s^{2} v^{3} + 5 \cdot 625 s v^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.23

Multiply $5$ by $625$ .

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} - 1250 s^{2} v^{3} + 3125 s v^{4} + 1 {(s)}^{0} {(- 5 v)}^{5}$

Step 4.24

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

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} - 1250 s^{2} v^{3} + 3125 s v^{4} + {(s)}^{0} {(- 5 v)}^{5}$

Step 4.25

Anything raised to $0$ is $1$ .

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} - 1250 s^{2} v^{3} + 3125 s v^{4} + 1 {(- 5 v)}^{5}$

Step 4.26

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

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} - 1250 s^{2} v^{3} + 3125 s v^{4} + {(- 5 v)}^{5}$

Step 4.27

Apply the product rule to $- 5 v$ .

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} - 1250 s^{2} v^{3} + 3125 s v^{4} + {(- 5)}^{5} v^{5}$

Step 4.28

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

$s^{5} - 25 s^{4} v + 250 s^{3} v^{2} - 1250 s^{2} v^{3} + 3125 s v^{4} - 3125 v^{5}$