Solve x^3+x^2+x^2+x+x^2+x+1

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

Evaluate

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

Solution Steps

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

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

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

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

Combine x and x to get 2x.

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

Differentiate w.r.t. x

$$3x^{2}+6x+2$$

Steps Using Derivative Rule for Sum

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

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

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

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

Combine x and x to get 2x.

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

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}.

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

Subtract 1 from 3.

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

Multiply 2 times 3.

$$3x^{2}+6x^{2-1}+2x^{1-1}$$

Subtract 1 from 2.

$$3x^{2}+6x^{1}+2x^{1-1}$$

Subtract 1 from 1.

$$3x^{2}+6x^{1}+2x^{0}$$

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

$$3x^{2}+6x+2x^{0}$$

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

$$3x^{2}+6x+2\times 1$$

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

$$3x^{2}+6x+2$$