Solve -3a^3+2a^2-4a+19a+7-2a^2-a^3

$$- 3 a ^ { 3 } + 2 a ^ { 2 } - 4 a + 19 a + 7 - 2 a ^ { 2 } - a ^ { 3 }$$

Evaluate

$$7+15a-4a^{3}$$

Solution Steps

Combine -4a and 19a to get 15a.

$$-3a^{3}+2a^{2}+15a+7-2a^{2}-a^{3}$$

Combine 2a^{2} and -2a^{2} to get 0.

$$-3a^{3}+15a+7-a^{3}$$

Combine -3a^{3} and -a^{3} to get -4a^{3}.

$$-4a^{3}+15a+7$$

Factor

$$\left(2a+1\right)\left(7+a-2a^{2}\right)$$

Solution Steps

Multiply and combine like terms.

$$-4a^{3}+15a+7$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 7 and q divides the leading coefficient -4. One such root is -\frac{1}{2}. Factor the polynomial by dividing it by 2a+1. Polynomial -2a^{2}+a+7 is not factored since it does not have any rational roots.

$$\left(2a+1\right)\left(-2a^{2}+a+7\right)$$