Zero Product Property
The "Zero Product Property" says that:
If a × b = 0 then a = 0 or b = 0
(or both a=0 and b=0)
It can help us solve equations:
Example: Solve (x−5)(x−3) = 0
The "Zero Product Property" says:
If (x−5)(x−3) = 0 then (x−5) = 0 or (x−3) = 0
Now we just solve each of those:
For (x−5) = 0 we get x = 5
For (x−3) = 0 we get x = 3
And the solutions are:
x = 5, or x = 3
Here it is on a graph:
y=0 when x=3 or x=5
Standard Form of an Equation
Sometimes we can solve an equation by putting it into Standard Form and then using the Zero Product Property:
The "Standard Form" of an equation is:
(some expression) = 0
In other words, "= 0" is on the right, and everything else is on the left.
Example: Put x2 = 7 into Standard Form
Answer:
x2 − 7 = 0
Standard Form and the Zero Product Property
So let's try it out:
Example: Solve 5(x+3) = 5x(x+3)
It is tempting to divide by (x+3), but that is dividing by zero when x = −3
So instead we can use "Standard Form":
5(x+3) − 5x(x+3) = 0
Which can be simplified to:
(5−5x)(x+3) = 0
5(1−x)(x+3) = 0
Then the "Zero Product Property" says:
(1−x) = 0, or (x+3) = 0
And the solutions are:
x = 1, or x = −3
And another example:
Example: Solve x3 = 25x
It is tempting to divide by x, but that is dividing by zero when x = 0
So let's use Standard Form and the Zero Product Property.
Bring all to the left hand side:
x3 − 25x = 0
Factor out x:
x(x2 − 25) = 0
x2 − 25 is a difference of squares, and can be factored into (x − 5)(x + 5):
x(x − 5)(x + 5) = 0
Now we can see three possible ways it could end up as zero:
x = 0, or x = 5, or x = −5