Solve (x^2+sqrt{5}i)(x^2-sqrt{5}i)(x^2+sqrt{5})(x^2-sqrt{5})

$$( x ^ { 2 } + \sqrt { 5 } i ) ( x ^ { 2 } - \sqrt { 5 } i ) ( x ^ { 2 } + \sqrt { 5 } ) ( x ^ { 2 } - \sqrt { 5 } )$$

Evaluate

$$x^{8}-25$$

Solution Steps

Use the distributive property to multiply x^{2}+i\sqrt{5} by x^{2}-i\sqrt{5}.

$$\left(x^{2}\left(x^{2}-i\sqrt{5}\right)+i\sqrt{5}\left(x^{2}-i\sqrt{5}\right)\right)\left(x^{2}+\sqrt{5}\right)\left(x^{2}-\sqrt{5}\right)$$

Use the distributive property to multiply x^{2}\left(x^{2}-i\sqrt{5}\right)+i\sqrt{5}\left(x^{2}-i\sqrt{5}\right) by x^{2}+\sqrt{5} and combine like terms.

$$\left(\left(x^{2}-i\sqrt{5}\right)x^{4}+\left(1+i\right)x^{2}\left(x^{2}-i\sqrt{5}\right)\sqrt{5}+i\left(x^{2}-i\sqrt{5}\right)\left(\sqrt{5}\right)^{2}\right)\left(x^{2}-\sqrt{5}\right)$$

The square of \sqrt{5} is 5.

$$\left(\left(x^{2}-i\sqrt{5}\right)x^{4}+\left(1+i\right)x^{2}\left(x^{2}-i\sqrt{5}\right)\sqrt{5}+i\left(x^{2}-i\sqrt{5}\right)\times 5\right)\left(x^{2}-\sqrt{5}\right)$$

Multiply i and 5 to get 5i.

$$\left(\left(x^{2}-i\sqrt{5}\right)x^{4}+\left(1+i\right)x^{2}\left(x^{2}-i\sqrt{5}\right)\sqrt{5}+5i\left(x^{2}-i\sqrt{5}\right)\right)\left(x^{2}-\sqrt{5}\right)$$

Use the distributive property to multiply \left(x^{2}-i\sqrt{5}\right)x^{4}+\left(1+i\right)x^{2}\left(x^{2}-i\sqrt{5}\right)\sqrt{5}+5i\left(x^{2}-i\sqrt{5}\right) by x^{2}-\sqrt{5} and combine like terms.

$$\left(x^{2}-i\sqrt{5}\right)x^{6}+i\left(x^{2}-i\sqrt{5}\right)x^{4}\sqrt{5}+\left(-1-i\right)\left(x^{2}-i\sqrt{5}\right)x^{2}\left(\sqrt{5}\right)^{2}+5i\left(x^{2}-i\sqrt{5}\right)x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

The square of \sqrt{5} is 5.

$$\left(x^{2}-i\sqrt{5}\right)x^{6}+i\left(x^{2}-i\sqrt{5}\right)x^{4}\sqrt{5}+\left(-1-i\right)\left(x^{2}-i\sqrt{5}\right)x^{2}\times 5+5i\left(x^{2}-i\sqrt{5}\right)x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

Multiply -1-i and 5 to get -5-5i.

$$\left(x^{2}-i\sqrt{5}\right)x^{6}+i\left(x^{2}-i\sqrt{5}\right)x^{4}\sqrt{5}+\left(-5-5i\right)\left(x^{2}-i\sqrt{5}\right)x^{2}+5i\left(x^{2}-i\sqrt{5}\right)x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

Combine \left(-5-5i\right)\left(x^{2}-i\sqrt{5}\right)x^{2} and 5i\left(x^{2}-i\sqrt{5}\right)x^{2} to get -5\left(x^{2}-i\sqrt{5}\right)x^{2}.

$$\left(x^{2}-i\sqrt{5}\right)x^{6}+i\left(x^{2}-i\sqrt{5}\right)x^{4}\sqrt{5}-5\left(x^{2}-i\sqrt{5}\right)x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

Use the distributive property to multiply x^{2}-i\sqrt{5} by x^{6}.

$$x^{8}-i\sqrt{5}x^{6}+i\left(x^{2}-i\sqrt{5}\right)x^{4}\sqrt{5}-5\left(x^{2}-i\sqrt{5}\right)x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

Use the distributive property to multiply i by x^{2}-i\sqrt{5}.

$$x^{8}-i\sqrt{5}x^{6}+\left(ix^{2}+\sqrt{5}\right)x^{4}\sqrt{5}-5\left(x^{2}-i\sqrt{5}\right)x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

Use the distributive property to multiply ix^{2}+\sqrt{5} by x^{4}.

$$x^{8}-i\sqrt{5}x^{6}+\left(ix^{6}+\sqrt{5}x^{4}\right)\sqrt{5}-5\left(x^{2}-i\sqrt{5}\right)x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

Use the distributive property to multiply ix^{6}+\sqrt{5}x^{4} by \sqrt{5}.

$$x^{8}-i\sqrt{5}x^{6}+ix^{6}\sqrt{5}+x^{4}\left(\sqrt{5}\right)^{2}-5\left(x^{2}-i\sqrt{5}\right)x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

The square of \sqrt{5} is 5.

$$x^{8}-i\sqrt{5}x^{6}+ix^{6}\sqrt{5}+x^{4}\times 5-5\left(x^{2}-i\sqrt{5}\right)x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

Combine -i\sqrt{5}x^{6} and ix^{6}\sqrt{5} to get 0.

$$x^{8}+x^{4}\times 5-5\left(x^{2}-i\sqrt{5}\right)x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

Use the distributive property to multiply -5 by x^{2}-i\sqrt{5}.

$$x^{8}+x^{4}\times 5+\left(-5x^{2}+5i\sqrt{5}\right)x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

Use the distributive property to multiply -5x^{2}+5i\sqrt{5} by x^{2}.

$$x^{8}+x^{4}\times 5-5x^{4}+5i\sqrt{5}x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

Combine x^{4}\times 5 and -5x^{4} to get 0.

$$x^{8}+5i\sqrt{5}x^{2}-5i\left(x^{2}-i\sqrt{5}\right)\sqrt{5}$$

Use the distributive property to multiply -5i by x^{2}-i\sqrt{5}.

$$x^{8}+5i\sqrt{5}x^{2}+\left(-5ix^{2}-5\sqrt{5}\right)\sqrt{5}$$

Use the distributive property to multiply -5ix^{2}-5\sqrt{5} by \sqrt{5}.

$$x^{8}+5i\sqrt{5}x^{2}-5ix^{2}\sqrt{5}-5\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$x^{8}+5i\sqrt{5}x^{2}-5ix^{2}\sqrt{5}-5\times 5$$

Multiply -5 and 5 to get -25.

$$x^{8}+5i\sqrt{5}x^{2}-5ix^{2}\sqrt{5}-25$$

Combine 5i\sqrt{5}x^{2} and -5ix^{2}\sqrt{5} to get 0.

$$x^{8}-25$$