Solve (x^2-sqrt{3}x+1)(x^2+sqrt{3}x+1)

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

Evaluate

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

Solution Steps

Use the distributive property to multiply x^{2}-\sqrt{3}x+1 by x^{2}+\sqrt{3}x+1.

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

Use the distributive property to multiply x^{2}-\sqrt{3}x by x^{2}.

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

Use the distributive property to multiply x^{2}-\sqrt{3}x by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Multiply -1 and 3 to get -3.

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

Use the distributive property to multiply x^{2}\sqrt{3}-3x by x.

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

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

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

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

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

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

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

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

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

Differentiate w.r.t. x

$$4x^{3}-2x$$

Steps Using Derivative Rule for Sum

Use the distributive property to multiply x^{2}-\sqrt{3}x+1 by x^{2}+\sqrt{3}x+1.

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

Use the distributive property to multiply x^{2}-\sqrt{3}x by x^{2}.

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

Use the distributive property to multiply x^{2}-\sqrt{3}x by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Multiply -1 and 3 to get -3.

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

Use the distributive property to multiply x^{2}\sqrt{3}-3x by x.

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

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

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

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

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

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

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

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

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

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}.

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

Subtract 1 from 4.

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

Multiply 2 times -1.

$$4x^{3}-2x^{2-1}$$

Subtract 1 from 2.

$$4x^{3}-2x^{1}$$

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

$$4x^{3}-2x$$