Solve a^3-3a^2+4

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

Factor

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

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 1. One such root is 2. Factor the polynomial by dividing it by a-2.

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

Consider a^{2}-a-2. Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-2. To find p and q, set up a system to be solved.

$$p+q=-1 pq=1\left(-2\right)=-2$$

Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$p=-2 q=1$$

Rewrite a^{2}-a-2 as \left(a^{2}-2a\right)+\left(a-2\right).

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

Factor out a in a^{2}-2a.

$$a\left(a-2\right)+a-2$$

Factor out common term a-2 by using distributive property.

$$\left(a-2\right)\left(a+1\right)$$

Rewrite the complete factored expression.

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

Evaluate

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