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

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

Evaluate

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

Solution Steps

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

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

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

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

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

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

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

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

Use the distributive property to multiply x^{2}-2x+1 by x^{2}+2x+1 and combine like terms.

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

Use the distributive property to multiply x^{4}-2x^{2}+1 by x^{4}+2x^{2}+1 and combine like terms.

$$x^{8}-2x^{4}+1$$

Expand

$$x^{8}-2x^{4}+1$$

Solution Steps

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

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

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

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

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

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

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

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

Use the distributive property to multiply x^{2}-2x+1 by x^{2}+2x+1 and combine like terms.

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

Use the distributive property to multiply x^{4}-2x^{2}+1 by x^{4}+2x^{2}+1 and combine like terms.

$$x^{8}-2x^{4}+1$$