Evaluate (1-x^2)^2

${(1 - x^{2})}^{2}$

Step 1

Rewrite ${(1 - x^{2})}^{2}$ as $(1 - x^{2}) (1 - x^{2})$ .

$(1 - x^{2}) (1 - x^{2})$

Step 2

Expand $(1 - x^{2}) (1 - x^{2})$ using the FOIL Method.

Apply the distributive property.

$1 (1 - x^{2}) - x^{2} (1 - x^{2})$

Apply the distributive property.

$1 \cdot 1 + 1 (- x^{2}) - x^{2} (1 - x^{2})$

Apply the distributive property.

$1 \cdot 1 + 1 (- x^{2}) - x^{2} \cdot 1 - x^{2} (- x^{2})$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$1 + 1 (- x^{2}) - x^{2} \cdot 1 - x^{2} (- x^{2})$

Multiply $- x^{2}$ by $1$ .

$1 - x^{2} - x^{2} \cdot 1 - x^{2} (- x^{2})$

Multiply $- 1$ by $1$ .

$1 - x^{2} - x^{2} - x^{2} (- x^{2})$

Rewrite using the commutative property of multiplication.

$1 - x^{2} - x^{2} - 1 \cdot - 1 x^{2} x^{2}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$1 - x^{2} - x^{2} - 1 \cdot - 1 (x^{2} x^{2})$

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

$1 - x^{2} - x^{2} - 1 \cdot - 1 x^{2 + 2}$

Add $2$ and $2$ .

$1 - x^{2} - x^{2} - 1 \cdot - 1 x^{4}$

Multiply $- 1$ by $- 1$ .

$1 - x^{2} - x^{2} + 1 x^{4}$

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

$1 - x^{2} - x^{2} + x^{4}$

Subtract $x^{2}$ from $- x^{2}$ .

$1 - 2 x^{2} + x^{4}$