Solve a^2-a^3+a^2-1

$$a ^ { 2 } - a ^ { 3 } + a ^ { 2 } - 1$$

Factor

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

Solution Steps

Multiply and combine like terms.

$$-a^{3}+2a^{2}-1$$

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

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

Evaluate

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

Solution Steps

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

$$2a^{2}-a^{3}-1$$