Solve (-2x^3+5x^2-5x+7)+(7x^3+10x^2-10x-3)

$$( - 2 x ^ { 3 } + 5 x ^ { 2 } - 5 x + 7 ) + ( 7 x ^ { 3 } + 10 x ^ { 2 } - 10 x - 3 )$$

Evaluate

$$5x^{3}+15x^{2}-15x+4$$

Solution Steps

Combine -2x^{3} and 7x^{3} to get 5x^{3}.

$$5x^{3}+5x^{2}-5x+7+10x^{2}-10x-3$$

Combine 5x^{2} and 10x^{2} to get 15x^{2}.

$$5x^{3}+15x^{2}-5x+7-10x-3$$

Combine -5x and -10x to get -15x.

$$5x^{3}+15x^{2}-15x+7-3$$

Subtract 3 from 7 to get 4.

$$5x^{3}+15x^{2}-15x+4$$

Differentiate w.r.t. x

$$15\left(x^{2}+2x-1\right)$$

Steps Using Derivative Rule for Sum

Combine -2x^{3} and 7x^{3} to get 5x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{3}+5x^{2}-5x+7+10x^{2}-10x-3)$$

Combine 5x^{2} and 10x^{2} to get 15x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{3}+15x^{2}-5x+7-10x-3)$$

Combine -5x and -10x to get -15x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{3}+15x^{2}-15x+7-3)$$

Subtract 3 from 7 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{3}+15x^{2}-15x+4)$$

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}.

$$3\times 5x^{3-1}+2\times 15x^{2-1}-15x^{1-1}$$

Multiply 3 times 5.

$$15x^{3-1}+2\times 15x^{2-1}-15x^{1-1}$$

Subtract 1 from 3.

$$15x^{2}+2\times 15x^{2-1}-15x^{1-1}$$

Multiply 2 times 15.

$$15x^{2}+30x^{2-1}-15x^{1-1}$$

Subtract 1 from 2.

$$15x^{2}+30x^{1}-15x^{1-1}$$

Subtract 1 from 1.

$$15x^{2}+30x^{1}-15x^{0}$$

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

$$15x^{2}+30x-15x^{0}$$

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

$$15x^{2}+30x-15$$