Solve a^3+2a^2-15a

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

Factor

$$a\left(a-3\right)\left(a+5\right)$$

Solution Steps

Factor out a.

$$a\left(a^{2}+2a-15\right)$$

Consider a^{2}+2a-15. Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-15. To find p and q, set up a system to be solved.

$$p+q=2 pq=1\left(-15\right)=-15$$

Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 2.

$$p=-3 q=5$$

Rewrite a^{2}+2a-15 as \left(a^{2}-3a\right)+\left(5a-15\right).

$$\left(a^{2}-3a\right)+\left(5a-15\right)$$

Factor out a in the first and 5 in the second group.

$$a\left(a-3\right)+5\left(a-3\right)$$

Factor out common term a-3 by using distributive property.

$$\left(a-3\right)\left(a+5\right)$$

Rewrite the complete factored expression.

$$a\left(a-3\right)\left(a+5\right)$$

Evaluate

$$a\left(a-3\right)\left(a+5\right)$$