Solve a^3-20a^2+19a

$$a ^ { 3 } - 20 a ^ { 2 } + 19 a$$

Factor

$$a\left(a-19\right)\left(a-1\right)$$

Solution Steps

Factor out a.

$$a\left(a^{2}-20a+19\right)$$

Consider a^{2}-20a+19. Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa+19. To find p and q, set up a system to be solved.

$$p+q=-20 pq=1\times 19=19$$

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

$$p=-19 q=-1$$

Rewrite a^{2}-20a+19 as \left(a^{2}-19a\right)+\left(-a+19\right).

$$\left(a^{2}-19a\right)+\left(-a+19\right)$$

Factor out a in the first and -1 in the second group.

$$a\left(a-19\right)-\left(a-19\right)$$

Factor out common term a-19 by using distributive property.

$$\left(a-19\right)\left(a-1\right)$$

Rewrite the complete factored expression.

$$a\left(a-19\right)\left(a-1\right)$$

Evaluate

$$a\left(a-19\right)\left(a-1\right)$$