Solve -ab+2a^2b-a^3b

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

Factor

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

Solution Steps

Factor out ab.

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

Consider -1+2a-a^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

Factor the expression by grouping. First, the expression needs to be rewritten as -a^{2}+pa+qa-1. To find p and q, set up a system to be solved.

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

Since pq is positive, p and q have the same sign. Since p+q is positive, p and q are both positive. The only such pair is the system solution.

$$p=1 q=1$$

Rewrite -a^{2}+2a-1 as \left(-a^{2}+a\right)+\left(a-1\right).

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

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

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

Factor out common term a-1 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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