Solve (-2x^3+10x^2-3x+2)+(4x^3+7x^2-2x-4)

$$( - 2 x ^ { 3 } + 10 x ^ { 2 } - 3 x + 2 ) + ( 4 x ^ { 3 } + 7 x ^ { 2 } - 2 x - 4 )$$

Evaluate

$$\left(2x-1\right)\left(x^{2}+9x+2\right)$$

Solution Steps

Combine -2x^{3} and 4x^{3} to get 2x^{3}.

$$2x^{3}+10x^{2}-3x+2+7x^{2}-2x-4$$

Combine 10x^{2} and 7x^{2} to get 17x^{2}.

$$2x^{3}+17x^{2}-3x+2-2x-4$$

Combine -3x and -2x to get -5x.

$$2x^{3}+17x^{2}-5x+2-4$$

Subtract 4 from 2 to get -2.

$$2x^{3}+17x^{2}-5x-2$$

Factor

$$\left(2x-1\right)\left(x^{2}+9x+2\right)$$

Solution Steps

Multiply and combine like terms.

$$2x^{3}+17x^{2}-5x-2$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2 and q divides the leading coefficient 2. One such root is \frac{1}{2}. Factor the polynomial by dividing it by 2x-1. Polynomial x^{2}+9x+2 is not factored since it does not have any rational roots.

$$\left(2x-1\right)\left(x^{2}+9x+2\right)$$