Expand the Trigonometric Expression (y-2x)^4

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Rewrite using the commutative property of multiplication.

$y^{4} + 4 \cdot - 2 y^{3} x + 6 y^{2} {(- 2 x)}^{2} + 4 y {(- 2 x)}^{3} + {(- 2 x)}^{4}$

Multiply $4$ by $- 2$ .

$y^{4} - 8 y^{3} x + 6 y^{2} {(- 2 x)}^{2} + 4 y {(- 2 x)}^{3} + {(- 2 x)}^{4}$

Apply the product rule to $- 2 x$ .

$y^{4} - 8 y^{3} x + 6 y^{2} ({(- 2)}^{2} x^{2}) + 4 y {(- 2 x)}^{3} + {(- 2 x)}^{4}$

Rewrite using the commutative property of multiplication.

$y^{4} - 8 y^{3} x + 6 \cdot {(- 2)}^{2} y^{2} x^{2} + 4 y {(- 2 x)}^{3} + {(- 2 x)}^{4}$

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

$y^{4} - 8 y^{3} x + 6 \cdot 4 y^{2} x^{2} + 4 y {(- 2 x)}^{3} + {(- 2 x)}^{4}$

Multiply $6$ by $4$ .

$y^{4} - 8 y^{3} x + 24 y^{2} x^{2} + 4 y {(- 2 x)}^{3} + {(- 2 x)}^{4}$

Apply the product rule to $- 2 x$ .

$y^{4} - 8 y^{3} x + 24 y^{2} x^{2} + 4 y ({(- 2)}^{3} x^{3}) + {(- 2 x)}^{4}$

Rewrite using the commutative property of multiplication.

$y^{4} - 8 y^{3} x + 24 y^{2} x^{2} + 4 \cdot {(- 2)}^{3} y x^{3} + {(- 2 x)}^{4}$

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

$y^{4} - 8 y^{3} x + 24 y^{2} x^{2} + 4 \cdot - 8 y x^{3} + {(- 2 x)}^{4}$

Multiply $4$ by $- 8$ .

$y^{4} - 8 y^{3} x + 24 y^{2} x^{2} - 32 y x^{3} + {(- 2 x)}^{4}$

Apply the product rule to $- 2 x$ .

$y^{4} - 8 y^{3} x + 24 y^{2} x^{2} - 32 y x^{3} + {(- 2)}^{4} x^{4}$

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

$y^{4} - 8 y^{3} x + 24 y^{2} x^{2} - 32 y x^{3} + 16 x^{4}$