Precalculus Examples

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

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 ${(3 x + 1)}^{5}$ , $n = 5$ 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 $a$ $3 x$ and $b$ $1$ into the expression.

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

Step 4

Simplify each term.

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Step 4.1

Multiply $1$ by ${(1)}^{0}$ by adding the exponents.

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Step 4.1.1

Move ${(1)}^{0}$ .

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.1.3

Add $0$ and $1$ .

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

Step 4.2

Simplify $1^{1} {(3 x)}^{5}$ .

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

Step 4.3

Apply the product rule to $3 x$ .

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

Step 4.4

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

$243 x^{5} + 5 {(3 x)}^{4} {(1)}^{1} + 10 {(3 x)}^{3} {(1)}^{2} + 10 {(3 x)}^{2} {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.5

Apply the product rule to $3 x$ .

$243 x^{5} + 5 (3^{4} x^{4}) {(1)}^{1} + 10 {(3 x)}^{3} {(1)}^{2} + 10 {(3 x)}^{2} {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.6

Raise $3$ to the power of $4$ .

$243 x^{5} + 5 (81 x^{4}) {(1)}^{1} + 10 {(3 x)}^{3} {(1)}^{2} + 10 {(3 x)}^{2} {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.7

Multiply $81$ by $5$ .

$243 x^{5} + 405 x^{4} {(1)}^{1} + 10 {(3 x)}^{3} {(1)}^{2} + 10 {(3 x)}^{2} {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.8

Evaluate the exponent.

$243 x^{5} + 405 x^{4} \cdot 1 + 10 {(3 x)}^{3} {(1)}^{2} + 10 {(3 x)}^{2} {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.9

Multiply $405$ by $1$ .

$243 x^{5} + 405 x^{4} + 10 {(3 x)}^{3} {(1)}^{2} + 10 {(3 x)}^{2} {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.10

Apply the product rule to $3 x$ .

$243 x^{5} + 405 x^{4} + 10 (3^{3} x^{3}) {(1)}^{2} + 10 {(3 x)}^{2} {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.11

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

$243 x^{5} + 405 x^{4} + 10 (27 x^{3}) {(1)}^{2} + 10 {(3 x)}^{2} {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.12

Multiply $27$ by $10$ .

$243 x^{5} + 405 x^{4} + 270 x^{3} {(1)}^{2} + 10 {(3 x)}^{2} {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.13

One to any power is one.

$243 x^{5} + 405 x^{4} + 270 x^{3} \cdot 1 + 10 {(3 x)}^{2} {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.14

Multiply $270$ by $1$ .

$243 x^{5} + 405 x^{4} + 270 x^{3} + 10 {(3 x)}^{2} {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.15

Apply the product rule to $3 x$ .

$243 x^{5} + 405 x^{4} + 270 x^{3} + 10 (3^{2} x^{2}) {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.16

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

$243 x^{5} + 405 x^{4} + 270 x^{3} + 10 (9 x^{2}) {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.17

Multiply $9$ by $10$ .

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} {(1)}^{3} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.18

One to any power is one.

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} \cdot 1 + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.19

Multiply $90$ by $1$ .

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} + 5 {(3 x)}^{1} {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.20

Simplify.

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} + 5 (3 x) {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.21

Multiply $3$ by $5$ .

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} + 15 x {(1)}^{4} + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.22

One to any power is one.

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} + 15 x \cdot 1 + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.23

Multiply $15$ by $1$ .

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} + 15 x + 1 {(3 x)}^{0} {(1)}^{5}$

Step 4.24

Multiply $1$ by ${(1)}^{5}$ by adding the exponents.

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Step 4.24.1

Move ${(1)}^{5}$ .

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} + 15 x + {(1)}^{5} \cdot 1 {(3 x)}^{0}$

Step 4.24.2

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

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Step 4.24.2.1

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

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} + 15 x + {(1)}^{5} \cdot 1^{1} {(3 x)}^{0}$

Step 4.24.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} + 15 x + 1^{5 + 1} {(3 x)}^{0}$

Step 4.24.3

Add $5$ and $1$ .

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} + 15 x + 1^{6} {(3 x)}^{0}$

Step 4.25

Simplify $1^{6} {(3 x)}^{0}$ .

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} + 15 x + 1^{6}$

Step 4.26

One to any power is one.

$243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} + 15 x + 1$