Algebra Examples

${(4 x - 3)}^{4}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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

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 $1 - 4 - 6 - 4 - 1$ .

$1 a^{4} b^{0} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + 1 a^{0} b^{4}$

Step 3

Substitute the actual values of $a$ $4 x$ and $b$ $- 3$ into the expression.

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

Step 4

Simplify each term.

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Step 4.1

Multiply ${(4 x)}^{4}$ by $1$ .

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

Step 4.2

Apply the product rule to $4 x$ .

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

Step 4.3

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

$256 x^{4} {(- 3)}^{0} + 4 {(4 x)}^{3} {(- 3)}^{1} + 6 {(4 x)}^{2} {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.4

Anything raised to $0$ is $1$ .

$256 x^{4} \cdot 1 + 4 {(4 x)}^{3} {(- 3)}^{1} + 6 {(4 x)}^{2} {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.5

Multiply $256$ by $1$ .

$256 x^{4} + 4 {(4 x)}^{3} {(- 3)}^{1} + 6 {(4 x)}^{2} {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.6

Apply the product rule to $4 x$ .

$256 x^{4} + 4 (4^{3} x^{3}) {(- 3)}^{1} + 6 {(4 x)}^{2} {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.7

Multiply $4$ by $4^{3}$ by adding the exponents.

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Step 4.7.1

Move $4^{3}$ .

$256 x^{4} + 4^{3} \cdot 4 x^{3} {(- 3)}^{1} + 6 {(4 x)}^{2} {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.7.2

Multiply $4^{3}$ by $4$ .

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Step 4.7.2.1

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

$256 x^{4} + 4^{3} \cdot 4^{1} x^{3} {(- 3)}^{1} + 6 {(4 x)}^{2} {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.7.2.2

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

$256 x^{4} + 4^{3 + 1} x^{3} {(- 3)}^{1} + 6 {(4 x)}^{2} {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.7.3

Add $3$ and $1$ .

$256 x^{4} + 4^{4} x^{3} {(- 3)}^{1} + 6 {(4 x)}^{2} {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.8

Simplify $4^{4} x^{3} {(- 3)}^{1}$ .

$256 x^{4} + 4^{4} x^{3} \cdot - 3 + 6 {(4 x)}^{2} {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.9

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

$256 x^{4} + 256 x^{3} \cdot - 3 + 6 {(4 x)}^{2} {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.10

Multiply $- 3$ by $256$ .

$256 x^{4} - 768 x^{3} + 6 {(4 x)}^{2} {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.11

Apply the product rule to $4 x$ .

$256 x^{4} - 768 x^{3} + 6 (4^{2} x^{2}) {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.12

Raise $4$ to the power of $2$ .

$256 x^{4} - 768 x^{3} + 6 (16 x^{2}) {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.13

Multiply $16$ by $6$ .

$256 x^{4} - 768 x^{3} + 96 x^{2} {(- 3)}^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.14

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

$256 x^{4} - 768 x^{3} + 96 x^{2} \cdot 9 + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.15

Multiply $9$ by $96$ .

$256 x^{4} - 768 x^{3} + 864 x^{2} + 4 {(4 x)}^{1} {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.16

Simplify.

$256 x^{4} - 768 x^{3} + 864 x^{2} + 4 (4 x) {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.17

Multiply $4$ by $4$ .

$256 x^{4} - 768 x^{3} + 864 x^{2} + 16 x {(- 3)}^{3} + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.18

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

$256 x^{4} - 768 x^{3} + 864 x^{2} + 16 x \cdot - 27 + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.19

Multiply $- 27$ by $16$ .

$256 x^{4} - 768 x^{3} + 864 x^{2} - 432 x + 1 {(4 x)}^{0} {(- 3)}^{4}$

Step 4.20

Multiply ${(4 x)}^{0}$ by $1$ .

$256 x^{4} - 768 x^{3} + 864 x^{2} - 432 x + {(4 x)}^{0} {(- 3)}^{4}$

Step 4.21

Apply the product rule to $4 x$ .

$256 x^{4} - 768 x^{3} + 864 x^{2} - 432 x + 4^{0} x^{0} {(- 3)}^{4}$

Step 4.22

Anything raised to $0$ is $1$ .

$256 x^{4} - 768 x^{3} + 864 x^{2} - 432 x + 1 x^{0} {(- 3)}^{4}$

Step 4.23

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

$256 x^{4} - 768 x^{3} + 864 x^{2} - 432 x + x^{0} {(- 3)}^{4}$

Step 4.24

Anything raised to $0$ is $1$ .

$256 x^{4} - 768 x^{3} + 864 x^{2} - 432 x + 1 {(- 3)}^{4}$

Step 4.25

Multiply ${(- 3)}^{4}$ by $1$ .

$256 x^{4} - 768 x^{3} + 864 x^{2} - 432 x + {(- 3)}^{4}$

Step 4.26

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

$256 x^{4} - 768 x^{3} + 864 x^{2} - 432 x + 81$