Solve (x^2+sqrt{2}x+1)(x^2-sqrt{2}x+1)

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

Evaluate

$$x^{4}+1$$

Solution Steps

Use the distributive property to multiply x^{2}+\sqrt{2}x+1 by x^{2}-\sqrt{2}x+1.

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

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

$$x^{4}-\sqrt{2}x^{3}+x^{2}+\sqrt{2}x\left(x^{2}-\sqrt{2}x\right)+\sqrt{2}x+x^{2}-\sqrt{2}x+1$$

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

$$x^{4}-\sqrt{2}x^{3}+x^{2}+\sqrt{2}x^{3}-\left(\sqrt{2}\right)^{2}x^{2}+\sqrt{2}x+x^{2}-\sqrt{2}x+1$$

The square of \sqrt{2} is 2.

$$x^{4}-\sqrt{2}x^{3}+x^{2}+\sqrt{2}x^{3}-2x^{2}+\sqrt{2}x+x^{2}-\sqrt{2}x+1$$

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

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

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

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

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

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

Combine \sqrt{2}x and -\sqrt{2}x to get 0.

$$x^{4}+1$$

Differentiate w.r.t. x

$$4x^{3}$$

Steps Using Derivative Rule for Sum

Use the distributive property to multiply x^{2}+\sqrt{2}x+1 by x^{2}-\sqrt{2}x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}\left(x^{2}-\sqrt{2}x\right)+x^{2}+\sqrt{2}x\left(x^{2}-\sqrt{2}x\right)+\sqrt{2}x+x^{2}-\sqrt{2}x+1)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}-\sqrt{2}x^{3}+x^{2}+\sqrt{2}x\left(x^{2}-\sqrt{2}x\right)+\sqrt{2}x+x^{2}-\sqrt{2}x+1)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}-\sqrt{2}x^{3}+x^{2}+\sqrt{2}x^{3}-\left(\sqrt{2}\right)^{2}x^{2}+\sqrt{2}x+x^{2}-\sqrt{2}x+1)$$

The square of \sqrt{2} is 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}-\sqrt{2}x^{3}+x^{2}+\sqrt{2}x^{3}-2x^{2}+\sqrt{2}x+x^{2}-\sqrt{2}x+1)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}+x^{2}-2x^{2}+\sqrt{2}x+x^{2}-\sqrt{2}x+1)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}-x^{2}+\sqrt{2}x+x^{2}-\sqrt{2}x+1)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}+\sqrt{2}x-\sqrt{2}x+1)$$

Combine \sqrt{2}x and -\sqrt{2}x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}+1)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

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

Subtract 1 from 4.

$$4x^{3}$$