Solve (x^2-1)^2-(1-x^2)^2=

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

Evaluate

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Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-1\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-2x^{2}+1-\left(1-x^{2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-x^{2}\right)^{2}.

$$x^{4}-2x^{2}+1-\left(1-2x^{2}+\left(x^{2}\right)^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-2x^{2}+1-\left(1-2x^{2}+x^{4}\right)$$

To find the opposite of 1-2x^{2}+x^{4}, find the opposite of each term.

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

Subtract 1 from 1 to get 0.

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

Combine -2x^{2} and 2x^{2} to get 0.

$$x^{4}-x^{4}$$

Combine x^{4} and -x^{4} to get 0.

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Factor

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Solution Steps

The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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Consider 2x^{2}-2. Factor out 2.

$$2\left(x^{2}-1\right)$$

Consider x^{2}-1. Rewrite x^{2}-1 as x^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-1\right)\left(x+1\right)$$

Rewrite the complete factored expression. Simplify.

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