Precalculus Examples

${(2 x + 5)}^{3}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

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 ${(2 x + 5)}^{3}$ , $n = 3$ so the coefficients of the expansion will correspond with line $4$ .

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 $1 - 3 - 3 - 1$ .

$1 a^{3} b^{0} + 3 a^{2} b + 3 a b^{2} + 1 a^{0} b^{3}$

Step 3

Substitute the actual values of $a$ $2 x$ and $b$ $5$ into the expression.

$1 {(2 x)}^{3} {(5)}^{0} + 3 {(2 x)}^{2} {(5)}^{1} + 3 {(2 x)}^{1} {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(2 x)}^{3}$ by $1$ .

${(2 x)}^{3} {(5)}^{0} + 3 {(2 x)}^{2} {(5)}^{1} + 3 {(2 x)}^{1} {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.2

Apply the product rule to $2 x$ .

$2^{3} x^{3} {(5)}^{0} + 3 {(2 x)}^{2} {(5)}^{1} + 3 {(2 x)}^{1} {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.3

Raise $2$ to the power of $3$ .

$8 x^{3} {(5)}^{0} + 3 {(2 x)}^{2} {(5)}^{1} + 3 {(2 x)}^{1} {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.4

Anything raised to $0$ is $1$ .

$8 x^{3} \cdot 1 + 3 {(2 x)}^{2} {(5)}^{1} + 3 {(2 x)}^{1} {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.5

Multiply $8$ by $1$ .

$8 x^{3} + 3 {(2 x)}^{2} {(5)}^{1} + 3 {(2 x)}^{1} {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.6

Apply the product rule to $2 x$ .

$8 x^{3} + 3 (2^{2} x^{2}) {(5)}^{1} + 3 {(2 x)}^{1} {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.7

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

$8 x^{3} + 3 (4 x^{2}) {(5)}^{1} + 3 {(2 x)}^{1} {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.8

Multiply $4$ by $3$ .

$8 x^{3} + 12 x^{2} {(5)}^{1} + 3 {(2 x)}^{1} {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.9

Evaluate the exponent.

$8 x^{3} + 12 x^{2} \cdot 5 + 3 {(2 x)}^{1} {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.10

Multiply $5$ by $12$ .

$8 x^{3} + 60 x^{2} + 3 {(2 x)}^{1} {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.11

Simplify.

$8 x^{3} + 60 x^{2} + 3 (2 x) {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.12

Multiply $2$ by $3$ .

$8 x^{3} + 60 x^{2} + 6 x {(5)}^{2} + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.13

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

$8 x^{3} + 60 x^{2} + 6 x \cdot 25 + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.14

Multiply $25$ by $6$ .

$8 x^{3} + 60 x^{2} + 150 x + 1 {(2 x)}^{0} {(5)}^{3}$

Step 4.15

Multiply ${(2 x)}^{0}$ by $1$ .

$8 x^{3} + 60 x^{2} + 150 x + {(2 x)}^{0} {(5)}^{3}$

Step 4.16

Apply the product rule to $2 x$ .

$8 x^{3} + 60 x^{2} + 150 x + 2^{0} x^{0} {(5)}^{3}$

Step 4.17

Anything raised to $0$ is $1$ .

$8 x^{3} + 60 x^{2} + 150 x + 1 x^{0} {(5)}^{3}$

Step 4.18

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

$8 x^{3} + 60 x^{2} + 150 x + x^{0} {(5)}^{3}$

Step 4.19

Anything raised to $0$ is $1$ .

$8 x^{3} + 60 x^{2} + 150 x + 1 {(5)}^{3}$

Step 4.20

Multiply ${(5)}^{3}$ by $1$ .

$8 x^{3} + 60 x^{2} + 150 x + {(5)}^{3}$

Step 4.21

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

$8 x^{3} + 60 x^{2} + 150 x + 125$