Solving Equations by Factoring
The Zero Factor Property
The equation ab = 0 indicates that the product of two unknown numbers is 0. But
the product of two real numbers is zero only when one or the other of the numbers
is 0. So even though we do not know exactly the values of a and b from ab = 0, we
do know that a = 0 or b = 0. This idea is called the zero factor property.
Zero Factor Property
The equation ab = 0 is equivalent to the compound equation
a = 0 or b = 0.
The next example shows how to use the zero factor property to solve an equation
in one variable.
Example 1
Using the zero factor property
Solve x^{2} + x  12 = 0.
Solution
We factor the lefthand side of the equation to get a product of two factors that are
equal to 0. Then we write an equivalent equation using the zero factor property.

x^{2} + x  12 = 0 



(x + 4)(x  3) = 0 

Factor the lefthand side. 
x + 4 = 0 
or 
x  3 = 0 

Zero factor property 
x = 4 
or 
x = 3 

Solve each part of the compound equation. 
Check that both 4 and 3 satisfy x^{2} + x  12 = 0. If x = 4, we get
(4)^{2} + (4)  12 = 16  4  12 = 0.
If x = 3, we get
(3)^{2} + 3  12 = 9 + 3  12 = 0.
So the solution set is {4, 3}.
The zero factor property is used only in solving polynomial equations that have
zero on one side and a polynomial that can be factored on the other side. The polynomials
that we factored most often were the quadratic polynomials. The equations that
we will solve most often using the zero factor property will be quadratic equations.
