Solve y^3-5y^2-24y

$$y ^ { 3 } - 5 y ^ { 2 } - 24 y$$

Factor

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

Solution Steps

Factor out y.

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

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

$$a+b=-5 ab=1\left(-24\right)=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b 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.

$$a=-8 b=3$$

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

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

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

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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