Finite Math Examples

${(3 x - 2)}^{4}$

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 ${(3 x - 2)}^{4}$ , $n = 4$ 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 $a$ $3 x$ and $b$ $- 2$ into the expression.

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

Apply the product rule to $3 x$ .

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

Step 4.3

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

$81 x^{4} {(- 2)}^{0} + 4 {(3 x)}^{3} {(- 2)}^{1} + 6 {(3 x)}^{2} {(- 2)}^{2} + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.4

Anything raised to $0$ is $1$ .

$81 x^{4} \cdot 1 + 4 {(3 x)}^{3} {(- 2)}^{1} + 6 {(3 x)}^{2} {(- 2)}^{2} + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.5

Multiply $81$ by $1$ .

$81 x^{4} + 4 {(3 x)}^{3} {(- 2)}^{1} + 6 {(3 x)}^{2} {(- 2)}^{2} + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.6

Apply the product rule to $3 x$ .

$81 x^{4} + 4 (3^{3} x^{3}) {(- 2)}^{1} + 6 {(3 x)}^{2} {(- 2)}^{2} + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.7

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

$81 x^{4} + 4 (27 x^{3}) {(- 2)}^{1} + 6 {(3 x)}^{2} {(- 2)}^{2} + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.8

Multiply $27$ by $4$ .

$81 x^{4} + 108 x^{3} {(- 2)}^{1} + 6 {(3 x)}^{2} {(- 2)}^{2} + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.9

Evaluate the exponent.

$81 x^{4} + 108 x^{3} \cdot - 2 + 6 {(3 x)}^{2} {(- 2)}^{2} + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.10

Multiply $- 2$ by $108$ .

$81 x^{4} - 216 x^{3} + 6 {(3 x)}^{2} {(- 2)}^{2} + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.11

Apply the product rule to $3 x$ .

$81 x^{4} - 216 x^{3} + 6 (3^{2} x^{2}) {(- 2)}^{2} + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.12

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

$81 x^{4} - 216 x^{3} + 6 (9 x^{2}) {(- 2)}^{2} + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.13

Multiply $9$ by $6$ .

$81 x^{4} - 216 x^{3} + 54 x^{2} {(- 2)}^{2} + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.14

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

$81 x^{4} - 216 x^{3} + 54 x^{2} \cdot 4 + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.15

Multiply $4$ by $54$ .

$81 x^{4} - 216 x^{3} + 216 x^{2} + 4 {(3 x)}^{1} {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.16

Simplify.

$81 x^{4} - 216 x^{3} + 216 x^{2} + 4 (3 x) {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.17

Multiply $3$ by $4$ .

$81 x^{4} - 216 x^{3} + 216 x^{2} + 12 x {(- 2)}^{3} + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.18

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

$81 x^{4} - 216 x^{3} + 216 x^{2} + 12 x \cdot - 8 + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.19

Multiply $- 8$ by $12$ .

$81 x^{4} - 216 x^{3} + 216 x^{2} - 96 x + 1 {(3 x)}^{0} {(- 2)}^{4}$

Step 4.20

Multiply ${(3 x)}^{0}$ by $1$ .

$81 x^{4} - 216 x^{3} + 216 x^{2} - 96 x + {(3 x)}^{0} {(- 2)}^{4}$

Step 4.21

Apply the product rule to $3 x$ .

$81 x^{4} - 216 x^{3} + 216 x^{2} - 96 x + 3^{0} x^{0} {(- 2)}^{4}$

Step 4.22

Anything raised to $0$ is $1$ .

$81 x^{4} - 216 x^{3} + 216 x^{2} - 96 x + 1 x^{0} {(- 2)}^{4}$

Step 4.23

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

$81 x^{4} - 216 x^{3} + 216 x^{2} - 96 x + x^{0} {(- 2)}^{4}$

Step 4.24

Anything raised to $0$ is $1$ .

$81 x^{4} - 216 x^{3} + 216 x^{2} - 96 x + 1 {(- 2)}^{4}$

Step 4.25

Multiply ${(- 2)}^{4}$ by $1$ .

$81 x^{4} - 216 x^{3} + 216 x^{2} - 96 x + {(- 2)}^{4}$

Step 4.26

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

$81 x^{4} - 216 x^{3} + 216 x^{2} - 96 x + 16$