Trigonometry Examples

${(x - 2)}^{8}$

Step 1

Use the Binomial Theorem.

$x^{8} + 8 x^{7} \cdot - 2 + 28 x^{6} {(- 2)}^{2} + 56 x^{5} {(- 2)}^{3} + 70 x^{4} {(- 2)}^{4} + 56 x^{3} {(- 2)}^{5} + 28 x^{2} {(- 2)}^{6} + 8 x {(- 2)}^{7} + {(- 2)}^{8}$

Step 2

Simplify each term.

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Step 2.1

Multiply $- 2$ by $8$ .

$x^{8} - 16 x^{7} + 28 x^{6} {(- 2)}^{2} + 56 x^{5} {(- 2)}^{3} + 70 x^{4} {(- 2)}^{4} + 56 x^{3} {(- 2)}^{5} + 28 x^{2} {(- 2)}^{6} + 8 x {(- 2)}^{7} + {(- 2)}^{8}$

Step 2.2

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

$x^{8} - 16 x^{7} + 28 x^{6} \cdot 4 + 56 x^{5} {(- 2)}^{3} + 70 x^{4} {(- 2)}^{4} + 56 x^{3} {(- 2)}^{5} + 28 x^{2} {(- 2)}^{6} + 8 x {(- 2)}^{7} + {(- 2)}^{8}$

Step 2.3

Multiply $4$ by $28$ .

$x^{8} - 16 x^{7} + 112 x^{6} + 56 x^{5} {(- 2)}^{3} + 70 x^{4} {(- 2)}^{4} + 56 x^{3} {(- 2)}^{5} + 28 x^{2} {(- 2)}^{6} + 8 x {(- 2)}^{7} + {(- 2)}^{8}$

Step 2.4

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

$x^{8} - 16 x^{7} + 112 x^{6} + 56 x^{5} \cdot - 8 + 70 x^{4} {(- 2)}^{4} + 56 x^{3} {(- 2)}^{5} + 28 x^{2} {(- 2)}^{6} + 8 x {(- 2)}^{7} + {(- 2)}^{8}$

Step 2.5

Multiply $- 8$ by $56$ .

$x^{8} - 16 x^{7} + 112 x^{6} - 448 x^{5} + 70 x^{4} {(- 2)}^{4} + 56 x^{3} {(- 2)}^{5} + 28 x^{2} {(- 2)}^{6} + 8 x {(- 2)}^{7} + {(- 2)}^{8}$

Step 2.6

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

$x^{8} - 16 x^{7} + 112 x^{6} - 448 x^{5} + 70 x^{4} \cdot 16 + 56 x^{3} {(- 2)}^{5} + 28 x^{2} {(- 2)}^{6} + 8 x {(- 2)}^{7} + {(- 2)}^{8}$

Step 2.7

Multiply $16$ by $70$ .

$x^{8} - 16 x^{7} + 112 x^{6} - 448 x^{5} + 1120 x^{4} + 56 x^{3} {(- 2)}^{5} + 28 x^{2} {(- 2)}^{6} + 8 x {(- 2)}^{7} + {(- 2)}^{8}$

Step 2.8

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

$x^{8} - 16 x^{7} + 112 x^{6} - 448 x^{5} + 1120 x^{4} + 56 x^{3} \cdot - 32 + 28 x^{2} {(- 2)}^{6} + 8 x {(- 2)}^{7} + {(- 2)}^{8}$

Step 2.9

Multiply $- 32$ by $56$ .

$x^{8} - 16 x^{7} + 112 x^{6} - 448 x^{5} + 1120 x^{4} - 1792 x^{3} + 28 x^{2} {(- 2)}^{6} + 8 x {(- 2)}^{7} + {(- 2)}^{8}$

Step 2.10

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

$x^{8} - 16 x^{7} + 112 x^{6} - 448 x^{5} + 1120 x^{4} - 1792 x^{3} + 28 x^{2} \cdot 64 + 8 x {(- 2)}^{7} + {(- 2)}^{8}$

Step 2.11

Multiply $64$ by $28$ .

$x^{8} - 16 x^{7} + 112 x^{6} - 448 x^{5} + 1120 x^{4} - 1792 x^{3} + 1792 x^{2} + 8 x {(- 2)}^{7} + {(- 2)}^{8}$

Step 2.12

Raise $- 2$ to the power of $7$ .

$x^{8} - 16 x^{7} + 112 x^{6} - 448 x^{5} + 1120 x^{4} - 1792 x^{3} + 1792 x^{2} + 8 x \cdot - 128 + {(- 2)}^{8}$

Step 2.13

Multiply $- 128$ by $8$ .

$x^{8} - 16 x^{7} + 112 x^{6} - 448 x^{5} + 1120 x^{4} - 1792 x^{3} + 1792 x^{2} - 1024 x + {(- 2)}^{8}$

Step 2.14

Raise $- 2$ to the power of $8$ .

$x^{8} - 16 x^{7} + 112 x^{6} - 448 x^{5} + 1120 x^{4} - 1792 x^{3} + 1792 x^{2} - 1024 x + 256$