In our Real Number Property Chart, we saw the Zero Property of Multiplication, which states that for any real number a, a • 0 = 0. This property can be expanded to state, that for any real numbers a and b, if a = 0 or b = 0, then a • b = 0.

Taking the "converse" of this expanded version (i.e., exchanging the "if" and the "then" in the statement), we will have the Zero Product Property.

Zero Product Property
The Zero Product Property states that if two or more quantities have a product of zero, then at least one (or more) of the quantities must be equal to zero.
If a • b = 0, the either a = 0, or b = 0 (or both a = 0 and b = 0).

While this property may seem to be simply a matter of common sense, the property carries a lot of weight mathematically. It is this property that allows us to solve polynomial equations by factoring, which can in turn allow us to easily graph polynomial functions.

NOTE: Expressions (polynomials) must be set equal to zero for this process to work.

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Consider: x2 - 2x - 15 = 0 (set quadratic trinomial expression = 0 creating an equation)
(x + 3)(x - 5) = 0 (factor the expression)
By the Zero Product Property, we know:
(x + 3) = 0    and/or    (x - 5) = 0
If x + 3 = 0, the x = -3
If x - 5 = 0, then x = 5

The zeros of this trinomial equation are -3 and 5.
You can check this result by substituting -3 and 5 back into the trinomial equation.

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Besides allowing the factors to find the zeros of an equation, this property can also be used in reverse to create the factors (thus an equation) using known zeros.

Consider: The zeros of a cubic equation (set = 0) are known to be
x = -1, x = 2 and x = 4.
By the Zero Product Property, we know:
x = -1 implies a factor of (x + 1) (where (x + 1) = 0)
x = 2 implies a factor of (x - 2) (where (x - 2) = 0)
x = 4 implies factor of (x - 4) (where (x - 4) = 0)
So, (x + 1)(x - 2)(x - 4)= 0
Thus, the equation: x3 - 5x2 + 2x + 8 = 0

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1
Find the solution (the zeros)
for this equation.
(x + 4)(2x - 1) = 0

Solution:
Using the Zero Product Property:
(x + 4) = 0
x + 4 = 0
x = -4
(2x - 1) = 0
2x - 1 = 0
2x = 1
x = ½
Zeros: {-4, ½}
Notice that a "zero" does not have to be an integer.
2
Find the solution (the zeros)
for this equation.
x2 + 5x - 14 = 0

Solution:
Factor first:
(x + 7)(x - 2) = 0
Using the Zero Product Property:
(x + 7) = 0
x + 7 = 0
x = -7
(x - 2) = 0
x - 2 = 0
x = 2
Zeros: {-7, 2}
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3
Find the solution (the zeros)
for this equation.
x2 - 121 = 0

Solution:
Factor first:
(x + 11)(x - 11) = 0
Using the Zero Product Property:
(x + 11) = 0
x + 11 = 0
x = -11
(x - 11) = 0
x - 11 = 0
x = 11
Zeros: {-11, 11}
4
Find the solution (the zeros)
for this equation.
2x2 - 20x + 48 = 0

Solution:
Factor first:
2(x2 - 10x + 24) = 0
2(x - 6)(x - 4) = 0
Using the Zero Product Property:
2 ≠ 0
(x - 6) = 0
x - 6 = 0
x = 6
(x - 4) = 0
x - 4 = 0
x = 4
Zeros: {4, 6}
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5
Find the solution (the zeros)
for this equation.
(x - 2)(x2 - 144) = 0

Solution:
More factoring is needed first:
(x - 2)(x + 12)(x - 12) = 0
Using the Zero Product Property:
(x - 2) = 0
x - 2 = 0
x = 2
(x + 12) = 0
x + 12 = 0
x = -12
(x - 12) = 0
x - 12 = 0
x = 12
Zeros: {-12, 2, 12} 		6
Write an equation that has zeros of -3, 0, and 10

Solution:
Working in reverse, we have:
x = -3, x = 0, and x = 10 as zeros.
(x + 3) = 0, x = 0 and (x - 10) = 0.
Using the Zero Property of Multiplication:
(x + 3) • x • (x - 10) = 0

Equation: x(x + 3)(x - 10) = 0
or in standard form:
x (x2 - 7x - 30) = 0
x3 - 7x2 - 30x = 0
Equation: x3 - 7x2 - 30x = 0

 

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The graphing calculator can prove to be helpful for checking your answer when factoring and graphing.
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For help with factoring on your calculator,
Click Here!

 

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