Zero Product Property
This page follows the JoeCMath video Zero Product Property in 2 Mins. The video explains why a product can equal zero only when at least one factor equals zero, then uses that idea to solve a factored equation.
Main Idea
The video starts with a product equal to zero:
\[a\cdot b=0.\]The important question is: how can multiplying two things produce zero? The video’s answer is the zero product property.
If the product is zero, then at least one factor must be zero.
Watch this section: introduction at 0:00.
The Zero Product Property
The video states the rule as:
\[\text{If }a\cdot b=0,\text{ then }a=0\text{ or }b=0.\]This does not mean both factors have to be zero. It means one factor could be zero, the other factor could be zero, or both could be zero.
The video briefly shows this with a concrete product like
\[5\cdot b=0.\]Since $5$ is not zero, the only way for the product to be zero is
\[b=0.\]Watch this section: the zero product property at 0:22.
The Factored Equation
The worked example in the video is already factored:
\[(x+3)(x-2)=0.\]That form matters. The zero product property is ready to use because the left side is written as a product and the right side is zero.
Watch this section: example that uses the zero product property at 0:40.
Set Each Factor Equal To Zero
Since
\[(x+3)(x-2)=0,\]the zero product property says that one of the factors must be zero. So the video separates the equation into two simpler equations:
\[x+3=0\quad\text{or}\quad x-2=0.\]Each equation comes from one factor of the original product.
Watch this section: setting the factors equal to zero at 0:40.
Solve The Two Small Equations
Now solve each small equation:
\[x+3=0\]so
\[x=-3.\]For the second factor,
\[x-2=0\]so
\[x=2.\]The solutions are
\[x=-3\quad\text{or}\quad x=2.\]Watch this section: solving the factor equations at 0:40.
Why Those Answers Work
The video checks the answers by putting each solution back into the original factored equation.
For $x=-3$,
\[(-3+3)(-3-2)=0\cdot(-5)=0.\]For $x=2$,
\[(2+3)(2-2)=5\cdot0=0.\]Each value makes one factor equal zero, so each value makes the whole product equal zero.
Watch this section: confirming the solutions at 0:40.
More Than Two Factors
Near the end, the video shows that the same idea works with more than two factors. For example, if
\[(2x-5)(x+2)(x-6)(x+10)=0,\]then each factor gives a possible equation:
\[2x-5=0,\quad x+2=0,\quad x-6=0,\quad x+10=0.\]The rule does not change. A product is zero when at least one factor is zero.
Watch this section: zero product property for more than two factors at 1:39.
Timestamp Guide
Related Algebra Work
The zero product property often appears after an equation has already been factored. It also connects naturally to polynomial sign work, where zeros split the number line into intervals. For that next step, use the companion guide for polynomial sign charts.
For the topic page around this post, start with Algebra equations.
The video description also points to the JoeCMath short math videos playlist: watch the playlist on YouTube.