Expand the Trigonometric Expression (x-4)^8

${(x - 4)}^{8}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Multiply $- 4$ by $8$ .

$x^{8} - 32 x^{7} + 28 x^{6} {(- 4)}^{2} + 56 x^{5} {(- 4)}^{3} + 70 x^{4} {(- 4)}^{4} + 56 x^{3} {(- 4)}^{5} + 28 x^{2} {(- 4)}^{6} + 8 x {(- 4)}^{7} + {(- 4)}^{8}$

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

$x^{8} - 32 x^{7} + 28 x^{6} \cdot 16 + 56 x^{5} {(- 4)}^{3} + 70 x^{4} {(- 4)}^{4} + 56 x^{3} {(- 4)}^{5} + 28 x^{2} {(- 4)}^{6} + 8 x {(- 4)}^{7} + {(- 4)}^{8}$

Multiply $16$ by $28$ .

$x^{8} - 32 x^{7} + 448 x^{6} + 56 x^{5} {(- 4)}^{3} + 70 x^{4} {(- 4)}^{4} + 56 x^{3} {(- 4)}^{5} + 28 x^{2} {(- 4)}^{6} + 8 x {(- 4)}^{7} + {(- 4)}^{8}$

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

$x^{8} - 32 x^{7} + 448 x^{6} + 56 x^{5} \cdot - 64 + 70 x^{4} {(- 4)}^{4} + 56 x^{3} {(- 4)}^{5} + 28 x^{2} {(- 4)}^{6} + 8 x {(- 4)}^{7} + {(- 4)}^{8}$

Multiply $- 64$ by $56$ .

$x^{8} - 32 x^{7} + 448 x^{6} - 3584 x^{5} + 70 x^{4} {(- 4)}^{4} + 56 x^{3} {(- 4)}^{5} + 28 x^{2} {(- 4)}^{6} + 8 x {(- 4)}^{7} + {(- 4)}^{8}$

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

$x^{8} - 32 x^{7} + 448 x^{6} - 3584 x^{5} + 70 x^{4} \cdot 256 + 56 x^{3} {(- 4)}^{5} + 28 x^{2} {(- 4)}^{6} + 8 x {(- 4)}^{7} + {(- 4)}^{8}$

Multiply $256$ by $70$ .

$x^{8} - 32 x^{7} + 448 x^{6} - 3584 x^{5} + 17920 x^{4} + 56 x^{3} {(- 4)}^{5} + 28 x^{2} {(- 4)}^{6} + 8 x {(- 4)}^{7} + {(- 4)}^{8}$

Raise $- 4$ to the power of $5$ .

$x^{8} - 32 x^{7} + 448 x^{6} - 3584 x^{5} + 17920 x^{4} + 56 x^{3} \cdot - 1024 + 28 x^{2} {(- 4)}^{6} + 8 x {(- 4)}^{7} + {(- 4)}^{8}$

Multiply $- 1024$ by $56$ .

$x^{8} - 32 x^{7} + 448 x^{6} - 3584 x^{5} + 17920 x^{4} - 57344 x^{3} + 28 x^{2} {(- 4)}^{6} + 8 x {(- 4)}^{7} + {(- 4)}^{8}$

Raise $- 4$ to the power of $6$ .

$x^{8} - 32 x^{7} + 448 x^{6} - 3584 x^{5} + 17920 x^{4} - 57344 x^{3} + 28 x^{2} \cdot 4096 + 8 x {(- 4)}^{7} + {(- 4)}^{8}$

Multiply $4096$ by $28$ .

$x^{8} - 32 x^{7} + 448 x^{6} - 3584 x^{5} + 17920 x^{4} - 57344 x^{3} + 114688 x^{2} + 8 x {(- 4)}^{7} + {(- 4)}^{8}$

Raise $- 4$ to the power of $7$ .

$x^{8} - 32 x^{7} + 448 x^{6} - 3584 x^{5} + 17920 x^{4} - 57344 x^{3} + 114688 x^{2} + 8 x \cdot - 16384 + {(- 4)}^{8}$

Multiply $- 16384$ by $8$ .

$x^{8} - 32 x^{7} + 448 x^{6} - 3584 x^{5} + 17920 x^{4} - 57344 x^{3} + 114688 x^{2} - 131072 x + {(- 4)}^{8}$

Raise $- 4$ to the power of $8$ .

$x^{8} - 32 x^{7} + 448 x^{6} - 3584 x^{5} + 17920 x^{4} - 57344 x^{3} + 114688 x^{2} - 131072 x + 65536$

Expand the Trigonometric Expression (x-4)^8

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