### Learning Outcomes

- Use factoring methods to factor polynomial equations

Not all of the techniques we use for solving linear equations will apply to solving polynomial equations. In this section, we will introduce a method for solving polynomial equations that combines factoring and the zero product principle.

## The Principle of Zero Products

What if we told you that we multiplied two numbers together and got an answer of zero? What could you say about the two numbers? Could they be [latex]2[/latex] and [latex]5[/latex]? Could they be [latex]9[/latex] and [latex]1[/latex]? No! When the result (answer) from multiplying two numbers is zero, that means that one of them *had *to be zero. This idea is called the zero product principle, and it is useful for solving polynomial equations that can be factored.

### Principle of Zero Products

The Principle of Zero Products states that if the product of two numbers is [latex]0[/latex], then at least one of the factors is [latex]0[/latex]. If [latex]ab=0[/latex], then either [latex]a=0[/latex] or [latex]b=0[/latex], or both *a* and *b* are [latex]0[/latex].

Let us start with a simple example. We will factor a GCF from a binomial and apply the principle of zero products to solve a polynomial equation.

### Example

Solve:

[latex]-t^2+t=0[/latex]

In the following video, we show two more examples of using both factoring and the principle of zero products to solve a polynomial equation.

In the next video, we show that you can use previously learned methods to factor a trinomial in order to solve a quadratic equation.

We all know that it is rare to be given an equation to solve that has zero on one side, so let us try another example.

### Example

Solve: [latex]s^2-4s=5[/latex]

We will work through one more example that is similar to the one above, except this example has fractions, yay!

### Example

Solve [latex]y^2-5=-\frac{7}{2}y+\frac{5}{2}[/latex]

In our last video, we show how to solve another quadratic equation that contains fractions.