Solve (x-4)(x-6-sqrt{2})(x-6+sqrt{2})

$$(x-4)(x-6- \sqrt{ 2 } )(x-6+ \sqrt{ 2 } )$$

Evaluate

$$\left(x-4\right)\left(x^{2}-12x+34\right)$$

Solution Steps

Apply the distributive property by multiplying each term of x-4 by each term of x-6-\sqrt{2}.

$$\left(x^{2}-6x-x\sqrt{2}-4x+24+4\sqrt{2}\right)\left(x-6+\sqrt{2}\right)$$

Combine -6x and -4x to get -10x.

$$\left(x^{2}-10x-x\sqrt{2}+24+4\sqrt{2}\right)\left(x-6+\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of x^{2}-10x-x\sqrt{2}+24+4\sqrt{2} by each term of x-6+\sqrt{2}.

$$x^{3}-6x^{2}+x^{2}\sqrt{2}-10x^{2}+60x-10x\sqrt{2}-\sqrt{2}x^{2}+6\sqrt{2}x-x\left(\sqrt{2}\right)^{2}+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

Combine -6x^{2} and -10x^{2} to get -16x^{2}.

$$x^{3}-16x^{2}+x^{2}\sqrt{2}+60x-10x\sqrt{2}-\sqrt{2}x^{2}+6\sqrt{2}x-x\left(\sqrt{2}\right)^{2}+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

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

$$x^{3}-16x^{2}+60x-10x\sqrt{2}+6\sqrt{2}x-x\left(\sqrt{2}\right)^{2}+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

Combine -10x\sqrt{2} and 6\sqrt{2}x to get -4x\sqrt{2}.

$$x^{3}-16x^{2}+60x-4x\sqrt{2}-x\left(\sqrt{2}\right)^{2}+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$x^{3}-16x^{2}+60x-4x\sqrt{2}-x\times 2+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

Multiply -1 and 2 to get -2.

$$x^{3}-16x^{2}+60x-4x\sqrt{2}-2x+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

Combine 60x and -2x to get 58x.

$$x^{3}-16x^{2}+58x-4x\sqrt{2}+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

Combine 58x and 24x to get 82x.

$$x^{3}-16x^{2}+82x-4x\sqrt{2}-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

Combine -4x\sqrt{2} and 4\sqrt{2}x to get 0.

$$x^{3}-16x^{2}+82x-144+24\sqrt{2}-24\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

Combine 24\sqrt{2} and -24\sqrt{2} to get 0.

$$x^{3}-16x^{2}+82x-144+4\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$x^{3}-16x^{2}+82x-144+4\times 2$$

Multiply 4 and 2 to get 8.

$$x^{3}-16x^{2}+82x-144+8$$

Add -144 and 8 to get -136.

$$x^{3}-16x^{2}+82x-136$$

Differentiate w.r.t. x

$$3x^{2}-32x+82$$

Steps Using Derivative Rule for Sum

Apply the distributive property by multiplying each term of x-4 by each term of x-6-\sqrt{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(x^{2}-6x-x\sqrt{2}-4x+24+4\sqrt{2}\right)\left(x-6+\sqrt{2}\right))$$

Combine -6x and -4x to get -10x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(x^{2}-10x-x\sqrt{2}+24+4\sqrt{2}\right)\left(x-6+\sqrt{2}\right))$$

Apply the distributive property by multiplying each term of x^{2}-10x-x\sqrt{2}+24+4\sqrt{2} by each term of x-6+\sqrt{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-6x^{2}+x^{2}\sqrt{2}-10x^{2}+60x-10x\sqrt{2}-\sqrt{2}x^{2}+6\sqrt{2}x-x\left(\sqrt{2}\right)^{2}+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2})$$

Combine -6x^{2} and -10x^{2} to get -16x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-16x^{2}+x^{2}\sqrt{2}+60x-10x\sqrt{2}-\sqrt{2}x^{2}+6\sqrt{2}x-x\left(\sqrt{2}\right)^{2}+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-16x^{2}+60x-10x\sqrt{2}+6\sqrt{2}x-x\left(\sqrt{2}\right)^{2}+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2})$$

Combine -10x\sqrt{2} and 6\sqrt{2}x to get -4x\sqrt{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-16x^{2}+60x-4x\sqrt{2}-x\left(\sqrt{2}\right)^{2}+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2})$$

The square of \sqrt{2} is 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-16x^{2}+60x-4x\sqrt{2}-x\times 2+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2})$$

Multiply -1 and 2 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-16x^{2}+60x-4x\sqrt{2}-2x+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2})$$

Combine 60x and -2x to get 58x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-16x^{2}+58x-4x\sqrt{2}+24x-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2})$$

Combine 58x and 24x to get 82x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-16x^{2}+82x-4x\sqrt{2}-144+24\sqrt{2}+4\sqrt{2}x-24\sqrt{2}+4\left(\sqrt{2}\right)^{2})$$

Combine -4x\sqrt{2} and 4\sqrt{2}x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-16x^{2}+82x-144+24\sqrt{2}-24\sqrt{2}+4\left(\sqrt{2}\right)^{2})$$

Combine 24\sqrt{2} and -24\sqrt{2} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-16x^{2}+82x-144+4\left(\sqrt{2}\right)^{2})$$

The square of \sqrt{2} is 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-16x^{2}+82x-144+4\times 2)$$

Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-16x^{2}+82x-144+8)$$

Add -144 and 8 to get -136.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-16x^{2}+82x-136)$$

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}.

$$3x^{3-1}+2\left(-16\right)x^{2-1}+82x^{1-1}$$

Subtract 1 from 3.

$$3x^{2}+2\left(-16\right)x^{2-1}+82x^{1-1}$$

Multiply 2 times -16.

$$3x^{2}-32x^{2-1}+82x^{1-1}$$

Subtract 1 from 2.

$$3x^{2}-32x^{1}+82x^{1-1}$$

Subtract 1 from 1.

$$3x^{2}-32x^{1}+82x^{0}$$

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

$$3x^{2}-32x+82x^{0}$$

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

$$3x^{2}-32x+82\times 1$$

For any term t, t\times 1=t and 1t=t.

$$3x^{2}-32x+82$$