Solve (x^2-2)(x-sqrt{2})(xsqrt{2}+2)

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

Evaluate

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

Solution Steps

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

$$\left(x^{3}-x^{2}\sqrt{2}-2x+2\sqrt{2}\right)\left(x\sqrt{2}+2\right)$$

Use the distributive property to multiply x^{3}-x^{2}\sqrt{2}-2x+2\sqrt{2} by x\sqrt{2}+2 and combine like terms.

$$\sqrt{2}x^{4}+2x^{3}-x^{3}\left(\sqrt{2}\right)^{2}-4\sqrt{2}x^{2}-4x+2x\left(\sqrt{2}\right)^{2}+4\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\sqrt{2}x^{4}+2x^{3}-x^{3}\times 2-4\sqrt{2}x^{2}-4x+2x\left(\sqrt{2}\right)^{2}+4\sqrt{2}$$

Multiply -1 and 2 to get -2.

$$\sqrt{2}x^{4}+2x^{3}-2x^{3}-4\sqrt{2}x^{2}-4x+2x\left(\sqrt{2}\right)^{2}+4\sqrt{2}$$

Combine 2x^{3} and -2x^{3} to get 0.

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

The square of \sqrt{2} is 2.

$$\sqrt{2}x^{4}-4\sqrt{2}x^{2}-4x+2x\times 2+4\sqrt{2}$$

Multiply 2 and 2 to get 4.

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

Combine -4x and 4x to get 0.

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

Factor

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