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

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

Evaluate

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

Solution Steps

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

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

Apply the distributive property by multiplying each term of x^{2}-x\sqrt{2} by each term of x+\sqrt{2}.

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

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

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

The square of \sqrt{2} is 2.

$$x^{3}-x\times 2$$

Multiply -1 and 2 to get -2.

$$x^{3}-2x$$

Differentiate w.r.t. x

$$3x^{2}-2$$

Steps Using Derivative Rule for Sum

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

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

Apply the distributive property by multiplying each term of x^{2}-x\sqrt{2} by each term of x+\sqrt{2}.

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

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

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

The square of \sqrt{2} is 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-x\times 2)$$

Multiply -1 and 2 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-2x)$$

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}.

$$3x^{3-1}-2x^{1-1}$$

Subtract 1 from 3.

$$3x^{2}-2x^{1-1}$$

Subtract 1 from 1.

$$3x^{2}-2x^{0}$$

For any term t except 0, t^{0}=1.

$$3x^{2}-2$$