Solve 3a^2b-15ab-72b

$$3 a ^ { 2 } b - 15 a b - 72 b$$

Factor

$$3b\left(a-8\right)\left(a+3\right)$$

Solution Steps

Factor out 3.

$$3\left(a^{2}b-5ab-24b\right)$$

Consider a^{2}b-5ab-24b. Factor out b.

$$b\left(a^{2}-5a-24\right)$$

Consider a^{2}-5a-24. Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-24. To find p and q, set up a system to be solved.

$$p+q=-5 pq=1\left(-24\right)=-24$$

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. List all such integer pairs that give product -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -5.

$$p=-8 q=3$$

Rewrite a^{2}-5a-24 as \left(a^{2}-8a\right)+\left(3a-24\right).

$$\left(a^{2}-8a\right)+\left(3a-24\right)$$

Factor out a in the first and 3 in the second group.

$$a\left(a-8\right)+3\left(a-8\right)$$

Factor out common term a-8 by using distributive property.

$$\left(a-8\right)\left(a+3\right)$$

Rewrite the complete factored expression.

$$3b\left(a-8\right)\left(a+3\right)$$

Evaluate

$$3b\left(a-8\right)\left(a+3\right)$$