Solve (3x^3-4x^2+5x-2)+(2x^3+6x^2-8x-5)

$$( 3 x ^ { 3 } - 4 x ^ { 2 } + 5 x - 2 ) + ( 2 x ^ { 3 } + 6 x ^ { 2 } - 8 x - 5 )$$

Evaluate

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

Solution Steps

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

$$5x^{3}-4x^{2}+5x-2+6x^{2}-8x-5$$

Combine -4x^{2} and 6x^{2} to get 2x^{2}.

$$5x^{3}+2x^{2}+5x-2-8x-5$$

Combine 5x and -8x to get -3x.

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

Subtract 5 from -2 to get -7.

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

Differentiate w.r.t. x

$$\left(3x-1\right)\left(5x+3\right)$$

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{3}-4x^{2}+5x-2+6x^{2}-8x-5)$$

Combine -4x^{2} and 6x^{2} to get 2x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{3}+2x^{2}+5x-2-8x-5)$$

Combine 5x and -8x to get -3x.

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

Subtract 5 from -2 to get -7.

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

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}.

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

Multiply 3 times 5.

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

Subtract 1 from 3.

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

Multiply 2 times 2.

$$15x^{2}+4x^{2-1}-3x^{1-1}$$

Subtract 1 from 2.

$$15x^{2}+4x^{1}-3x^{1-1}$$

Subtract 1 from 1.

$$15x^{2}+4x^{1}-3x^{0}$$

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

$$15x^{2}+4x-3x^{0}$$

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

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