Solve x^3y+6x^2y-27xy

$$x ^ { 3 } y + 6 x ^ { 2 } y - 27 x y$$

Factor

$$xy\left(x-3\right)\left(x+9\right)$$

Solution Steps

Factor out xy.

$$xy\left(x^{2}+6x-27\right)$$

Consider x^{2}+6x-27. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-27. To find a and b, set up a system to be solved.

$$a+b=6 ab=1\left(-27\right)=-27$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -27.

$$-1,27 -3,9$$

Calculate the sum for each pair.

$$-1+27=26 -3+9=6$$

The solution is the pair that gives sum 6.

$$a=-3 b=9$$

Rewrite x^{2}+6x-27 as \left(x^{2}-3x\right)+\left(9x-27\right).

$$\left(x^{2}-3x\right)+\left(9x-27\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x-3\right)+9\left(x-3\right)$$

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

$$\left(x-3\right)\left(x+9\right)$$

Rewrite the complete factored expression.

$$xy\left(x-3\right)\left(x+9\right)$$

Evaluate

$$xy\left(x-3\right)\left(x+9\right)$$