Notes: Zero Product Property
I. Zero Product Property
If a • b = 0 then
a=0,
b=0,
or both a and b equal 0.
Four Steps to Solving Using ZPP
Get Zero on one side of the equal sign
Factor the equation
Split the equation up
Solve each equation
Zero, Factor, Split, Solve
I. Basic Examples, GCF
1. Solve (x + 3)(x - 5) = 0
-Zero is on one side of the equal sign
-The equation is already factored
-Split each set of
parentheses into its
own equation
Solve each equation
x + 3 = 0 or x - 5 = 0
x = -3 or x = 5
{-3, 5}
2. Solve (2a + 4)(a + 7) = 0
-Zero is on one side of
the equal sign
-The equation is already
factored
-Split each set of
parentheses into its
own equation
Solve each equation
2a + 4 = 0 or a + 7 = 0
2a = -4 or a = -7
a = -2 or a = -7
{-2, -7}
3. Solve (3t + 5)(t - 3) = 0
-Zero is on one side of
the equal sign
-The equation is already
factored
-Split each set of
parentheses into its
own equation
3t + 5 = 0 or t - 3 = 0
3t = -5 or t = 3
t = -5/3
or t = 3
Solve each equation
{-5/3, 3}
Solve (y – 3)(2y + 6) = 0
1.
2.
3.
4.
{-3, 3}
{-3, 6}
{3, 6}
{3, -6}
III. Examples Involving Factoring
-Zero is on one side of
the equal sign
-Factor the equation
1. Solve x2 - 11x = 0
-Split each factor into
its own equation
x = 0 or x - 11 = 0
x(x - 11) = 0
x = 0 or x = 11
Solve each equation
{0, 11}
2. Solve -24a +144 = -a2
Get Zero on one side
of the equal sign
-Factor the equation
a2 - 24a + 144 = 0
-Split each set of parentheses
into its own equation. (In this
case it is the same equation
twice, so we only need to
solve for it once.)
Solve each equation
(a - 12)2 = 0
a - 12 = 0
a = 12
{12}
3. Solve
2
4m
+ 25 = 20m
4m2 - 20m + 25 = 0
(2m - 5)2 = 0
2m - 5 = 0
2m = 5
5
m=
2
5
  or 2.5
2
Zero
Factor
Split
Solve
4. Solve
3
x
+
2
2x
= 15x
x3 + 2x2 - 15x = 0
Zero
2
x(x + 2x - 15) = 0
Factor
Split
x(x + 5)(x - 3) = 0
Solve
x = 0 or x + 5 = 0 or x - 3 = 0
{0, -5, 3}
Ex 5:
x  3x  18
2
x  3x  18  0
2
 x  3  x  6   0
x3 0
x  3
x6  0
x6
Zero
Factor
Split
Solve
Ex 6:
x  x  4  5
x  4x  5
2
x  4x  5  0
2
 x  1 x  5  0
x 1  0
x 5  0
x  1
x5
You cannot
split until
you have
zero on one
side!!!
E 7:
x  3x  7   6
3x  7 x  6
2
3x  7 x  6  0
 x  33x  2  0
2
x  3  0 3x  2  0
x  3
3x  2
x2
3
Ex 8:
5 x  15 x  5 x  15  0
3
2
5( x  3 x  x  3)  0
3
2
3
2

5 ( x  3 x )( x  3)   0
2
5  x ( x  3)  1( x  3)   0
Zero
Factor
Split
Solve
5( x  1)( x  3)  0
5( x  1)( x  1)( x  3)  0
2
 x  1  0
 x  1  0
x  1
x 1
 x  3  0
x  3
Solve
1.
2.
3.
4.
{-8, 5}
{-5, 8}
{-8, -5}
{5, 8}
2
a
– 3a = 40
Solve
1.
2.
3.
4.
5.
3
4r
{-16, 4}
{-4, 16}
{0, 2}
{0, 4}
{-2, 0, 2}
Hint: The degree will
tell you how many
answers you have!
– 16r = 0