17Calculus Precalculus - Zero Product Rule
The Zero Product Rule (also called Zero Product Property) is a simple yet powerful rule that you will use a lot in calculus. Here is how it works.
If you have an equation like \( (a) (b) = 0 \), you can automatically assume that either \(a = 0 \) and/or \(b=0\). The reason this works is that when you multiply any number by zero, the result is always zero. For example, \( (2) 0 = 0 \) and \( 0 (621) = 0 \). Any number or variable multiplied by zero is always equal to zero.
This also works with more than one variable. For example, if you have \( (x) (y) (z) (a) = 0\), then you can write \( x=0 \), \(y=0\), \(z=0\) and/or \(a=0\). The key is that one or more of them have to be zero for this equation be true.
One common mistake students make is to assume this works when you have something other than zero on one side of the equation. For example, if you have \( (a) (b) = 1 \), you cannot assume that \( a=1\) or \( b=1\). This would hold if \( a=2\) and \( b=1/2\) neither of which are \(1\). So the key when using the zero product rule is that you need zero on one side of the equation.
Another thing to watch is if you have something other multiplication, i.e. \( (a) (b) + 1 = 0 \). Of course, you should know by now that you can't say \( a=0 \) and/or \(b=0\). The \( +1 \) term does not allow you to apply the zero product rule.
This rule is used mostly when solving for roots, as in this example.
Example
Find the roots (where the function crosses the x-axis) of \( f(x) = x^2 - 3x + 2 \).
Solution
When the function crosses the x-axis, the y-value is zero, so we need to solve \( x^2 - 3x +2 = 0 \). First, we factor.
\(
\begin{array}{rcl}
x^2 - 3x +2 & = & 0 \\
(x-1)(x-2) & = & 0
\end{array}
\)
Notice we have two expressions within parentheses multiplied together and we have zero on the other side of the equal sign. So we can apply the zero product rule.
\( x-1 = 0 \) or \( x-2 = 0 \)
Solving for \(x\) in both equations gives us
\( x=1 \) or \( x=2 \)
So we know that the roots of \( f(x) \) are \( x=1\) and \( x=2\).
Okay, time for some practice problems.
Practice
Instructions - - Unless otherwise instructed, use the zero product rule to solve these equations, i.e. find the x-values that make these equations true.
\( x(x-3) = 0 \)
Problem Statement |
|---|
Solve \( x(x-3) = 0 \) using the zero product rule.
Final Answer |
|---|
\(x = 0\) or \(x = 3\)
Problem Statement |
|---|
Solve \( x(x-3) = 0 \) using the zero product rule.
Solution |
|---|
2740 video
Final Answer |
|---|
\(x = 0\) or \(x = 3\) |
|
close solution
|
Log in to rate this practice problem. |
|---|
\( (x+2)(x-3) = 0 \)
Problem Statement |
|---|
Solve \( (x+2)(x-3) = 0 \) using the zero product rule.
Final Answer |
|---|
\(x = -2\) or \( x = 3 \)
Problem Statement |
|---|
Solve \( (x+2)(x-3) = 0 \) using the zero product rule.
Solution |
|---|
2741 video
Final Answer |
|---|
\(x = -2\) or \( x = 3 \) |
|
close solution
|
Log in to rate this practice problem. |
|---|
\( 4x(x+5)=0 \)
Problem Statement |
|---|
Solve \( 4x(x+5)=0 \) using the zero product rule.
Final Answer |
|---|
\( x = 0 \) or \( x = -5 \)
Problem Statement |
|---|
Solve \( 4x(x+5)=0 \) using the zero product rule.
Solution |
|---|
2742 video
Final Answer |
|---|
\( x = 0 \) or \( x = -5 \) |
|
close solution
|
Log in to rate this practice problem. |
|---|
\( (x-2)(x+7) = 0 \)
Problem Statement |
|---|
Solve \( (x-2)(x+7) = 0 \) using the zero product rule.
Final Answer |
|---|
\( x = 2 \) or \( x = -7 \)
Problem Statement |
|---|
Solve \( (x-2)(x+7) = 0 \) using the zero product rule.
Solution |
|---|
2743 video
Final Answer |
|---|
\( x = 2 \) or \( x = -7 \) |
|
close solution
|
Log in to rate this practice problem. |
|---|
\( (2x+3)(5x-1) = 0 \)
Problem Statement |
|---|
Solve \( (2x+3)(5x-1) = 0 \) using the zero product rule.
Final Answer |
|---|
\( x = -3/2 \) or \( x = 1/5 \)
Problem Statement |
|---|
Solve \( (2x+3)(5x-1) = 0 \) using the zero product rule.
Solution |
|---|
2744 video
Final Answer |
|---|
\( x = -3/2 \) or \( x = 1/5 \) |
|
close solution
|
Log in to rate this practice problem. |
|---|
\( x(x-1)(6x+11) = 0 \)
Problem Statement |
|---|
Solve \( x(x-1)(6x+11) = 0 \) using the zero product rule.
Final Answer |
|---|
\( x = 0 \) or \( x = -1 \) or \( x = -11/6 \)
Problem Statement |
|---|
Solve \( x(x-1)(6x+11) = 0 \) using the zero product rule.
Solution |
|---|
2745 video
Final Answer |
|---|
\( x = 0 \) or \( x = -1 \) or \( x = -11/6 \) |
|
close solution
|
Log in to rate this practice problem. |
|---|
Problem Statement |
|---|
\(-x+x^2=6\)
Solution |
|---|
2135 video
video by Krista King Math |
|---|
|
close solution
|
Log in to rate this practice problem. |
|---|
\( x^2 - x - 12 = 0 \)
Problem Statement |
|---|
Solve \( x^2 - x - 12 = 0 \) by factoring and applying the zero product rule.
Final Answer |
|---|
\( x = 4 \) or \( x = -3 \)
Problem Statement |
|---|
Solve \( x^2 - x - 12 = 0 \) by factoring and applying the zero product rule.
Solution |
|---|
2746 video
Final Answer |
|---|
\( x = 4 \) or \( x = -3 \) |
|
close solution
|
Log in to rate this practice problem. |
|---|
\( x^2 + 6x - 16 = 0 \)
Problem Statement |
|---|
Solve by \( x^2 + 6x - 16 = 0 \) factoring and applying the zero product rule.
Solution |
|---|
2747 video
|
close solution
|
Log in to rate this practice problem. |
|---|
\( x^2 - 9x = 0 \)
Problem Statement |
|---|
Solve \( x^2 - 9x = 0 \) by factoring and applying the zero product rule.
Final Answer |
|---|
\( x = 0 \) or \( x = 9 \)
Problem Statement |
|---|
Solve \( x^2 - 9x = 0 \) by factoring and applying the zero product rule.
Solution |
|---|
2748 video
Final Answer |
|---|
\( x = 0 \) or \( x = 9 \) |
|
close solution
|
Log in to rate this practice problem. |
|---|
\( x^2 - 3x - 10 = 0 \)
Problem Statement |
|---|
Solve \( x^2 - 3x - 10 = 0 \) by factoring and applying the zero product rule.
Final Answer |
|---|
\( x = 5 \) or \( x = -2 \)
Problem Statement |
|---|
Solve \( x^2 - 3x - 10 = 0 \) by factoring and applying the zero product rule.
Solution |
|---|
2749 video
Final Answer |
|---|
\( x = 5 \) or \( x = -2 \) |
|
close solution
|
Log in to rate this practice problem. |
|---|
\( x^2 - 36 = 0 \)
Problem Statement |
|---|
Solve \( x^2 - 36 = 0 \) by factoring and applying the zero product rule.
Final Answer |
|---|
\( x = 6 \) or \( x = -6 \)
Problem Statement |
|---|
Solve \( x^2 - 36 = 0 \) by factoring and applying the zero product rule.
Solution |
|---|
2750 video
Final Answer |
|---|
\( x = 6 \) or \( x = -6 \) |
|
close solution
|
Log in to rate this practice problem. |
|---|
\( 3x^2 + 4x - 7 = 0 \)
Problem Statement |
|---|
Solve \( 3x^2 + 4x - 7 = 0 \) by factoring and applying the zero product rule.
Final Answer |
|---|
\( x = 0 \) or \( x = -7/3 \)
Problem Statement |
|---|
Solve \( 3x^2 + 4x - 7 = 0 \) by factoring and applying the zero product rule.
Solution |
|---|
2751 video
Final Answer |
|---|
\( x = 0 \) or \( x = -7/3 \) |
|
close solution
|
Log in to rate this practice problem. |
|---|
zero product rule 17calculus youtube playlist
Really UNDERSTAND Precalculus
Topics You Need To Understand For This Page
algebra factoring |
Related Topics and Links
external links you may find helpful |
|---|
To bookmark this page and practice problems, log in to your account or set up a free account.
Calculus Topics Listed Alphabetically
Single Variable Calculus |
|---|
Multi-Variable Calculus |
|---|
Differential Equations Topics Listed Alphabetically
Precalculus Topics Listed Alphabetically
Search Practice Problems
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
|
| |
|
Help Keep 17Calculus Free |
|---|
The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free. |
