Solve a^3+2a^2-35a

$$a ^ { 3 } + 2 a ^ { 2 } - 35 a$$

Factor

$$a\left(a-5\right)\left(a+7\right)$$

Solution Steps

Factor out a.

$$a\left(a^{2}+2a-35\right)$$

Consider a^{2}+2a-35. Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-35. To find p and q, set up a system to be solved.

$$p+q=2 pq=1\left(-35\right)=-35$$

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 -35.

$$-1,35 -5,7$$

Calculate the sum for each pair.

$$-1+35=34 -5+7=2$$

The solution is the pair that gives sum 2.

$$p=-5 q=7$$

Rewrite a^{2}+2a-35 as \left(a^{2}-5a\right)+\left(7a-35\right).

$$\left(a^{2}-5a\right)+\left(7a-35\right)$$

Factor out a in the first and 7 in the second group.

$$a\left(a-5\right)+7\left(a-5\right)$$

Factor out common term a-5 by using distributive property.

$$\left(a-5\right)\left(a+7\right)$$

Rewrite the complete factored expression.

$$a\left(a-5\right)\left(a+7\right)$$

Evaluate

$$a\left(a-5\right)\left(a+7\right)$$