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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 x2 + x - 12 = 0.

Solution

We factor the left-hand 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.

  x2 + x - 12 = 0    
  (x + 4)(x - 3) = 0   Factor the left-hand 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 x2 + 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.

 

 
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