Precalculus Examples

${(x - 6)}^{5}$

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 ${(x - 6)}^{5}$ , $n = 5$ 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 $a$ $x$ and $b$ $- 6$ into the expression.

$1 {(x)}^{5} {(- 6)}^{0} + 5 {(x)}^{4} {(- 6)}^{1} + 10 {(x)}^{3} {(- 6)}^{2} + 10 {(x)}^{2} {(- 6)}^{3} + 5 {(x)}^{1} {(- 6)}^{4} + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4

Simplify each term.

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Step 4.1

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

${(x)}^{5} {(- 6)}^{0} + 5 {(x)}^{4} {(- 6)}^{1} + 10 {(x)}^{3} {(- 6)}^{2} + 10 {(x)}^{2} {(- 6)}^{3} + 5 {(x)}^{1} {(- 6)}^{4} + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4.2

Anything raised to $0$ is $1$ .

$x^{5} \cdot 1 + 5 {(x)}^{4} {(- 6)}^{1} + 10 {(x)}^{3} {(- 6)}^{2} + 10 {(x)}^{2} {(- 6)}^{3} + 5 {(x)}^{1} {(- 6)}^{4} + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4.3

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

$x^{5} + 5 {(x)}^{4} {(- 6)}^{1} + 10 {(x)}^{3} {(- 6)}^{2} + 10 {(x)}^{2} {(- 6)}^{3} + 5 {(x)}^{1} {(- 6)}^{4} + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4.4

Evaluate the exponent.

$x^{5} + 5 x^{4} \cdot - 6 + 10 {(x)}^{3} {(- 6)}^{2} + 10 {(x)}^{2} {(- 6)}^{3} + 5 {(x)}^{1} {(- 6)}^{4} + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4.5

Multiply $- 6$ by $5$ .

$x^{5} - 30 x^{4} + 10 {(x)}^{3} {(- 6)}^{2} + 10 {(x)}^{2} {(- 6)}^{3} + 5 {(x)}^{1} {(- 6)}^{4} + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4.6

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

$x^{5} - 30 x^{4} + 10 x^{3} \cdot 36 + 10 {(x)}^{2} {(- 6)}^{3} + 5 {(x)}^{1} {(- 6)}^{4} + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4.7

Multiply $36$ by $10$ .

$x^{5} - 30 x^{4} + 360 x^{3} + 10 {(x)}^{2} {(- 6)}^{3} + 5 {(x)}^{1} {(- 6)}^{4} + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4.8

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

$x^{5} - 30 x^{4} + 360 x^{3} + 10 x^{2} \cdot - 216 + 5 {(x)}^{1} {(- 6)}^{4} + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4.9

Multiply $- 216$ by $10$ .

$x^{5} - 30 x^{4} + 360 x^{3} - 2160 x^{2} + 5 {(x)}^{1} {(- 6)}^{4} + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4.10

Simplify.

$x^{5} - 30 x^{4} + 360 x^{3} - 2160 x^{2} + 5 x {(- 6)}^{4} + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4.11

Raise $- 6$ to the power of $4$ .

$x^{5} - 30 x^{4} + 360 x^{3} - 2160 x^{2} + 5 x \cdot 1296 + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4.12

Multiply $1296$ by $5$ .

$x^{5} - 30 x^{4} + 360 x^{3} - 2160 x^{2} + 6480 x + 1 {(x)}^{0} {(- 6)}^{5}$

Step 4.13

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

$x^{5} - 30 x^{4} + 360 x^{3} - 2160 x^{2} + 6480 x + {(x)}^{0} {(- 6)}^{5}$

Step 4.14

Anything raised to $0$ is $1$ .

$x^{5} - 30 x^{4} + 360 x^{3} - 2160 x^{2} + 6480 x + 1 {(- 6)}^{5}$

Step 4.15

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

$x^{5} - 30 x^{4} + 360 x^{3} - 2160 x^{2} + 6480 x + {(- 6)}^{5}$

Step 4.16

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

$x^{5} - 30 x^{4} + 360 x^{3} - 2160 x^{2} + 6480 x - 7776$