Solve y^3-4y^2-45y

$$y ^ { 3 } - 4 y ^ { 2 } - 45 y$$

Factor

$$y\left(y-9\right)\left(y+5\right)$$

Solution Steps

Factor out y.

$$y\left(y^{2}-4y-45\right)$$

Consider y^{2}-4y-45. Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by-45. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\left(-45\right)=-45$$

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 -45.

$$1,-45 3,-15 5,-9$$

Calculate the sum for each pair.

$$1-45=-44 3-15=-12 5-9=-4$$

The solution is the pair that gives sum -4.

$$a=-9 b=5$$

Rewrite y^{2}-4y-45 as \left(y^{2}-9y\right)+\left(5y-45\right).

$$\left(y^{2}-9y\right)+\left(5y-45\right)$$

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

$$y\left(y-9\right)+5\left(y-9\right)$$

Factor out common term y-9 by using distributive property.

$$\left(y-9\right)\left(y+5\right)$$

Rewrite the complete factored expression.

$$y\left(y-9\right)\left(y+5\right)$$

Evaluate

$$y\left(y-9\right)\left(y+5\right)$$