# 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 x^{2} = 7 into Standard Form

Answer:

x^{2} − 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 **x**^{3} = 25x

^{3}= 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:

x^{3} − 25x = 0

Factor out x:

x(x^{2} − 25) = 0

x^{2} − 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**