### Learning Objectives

- Factor the greatest common monomial out of a polynomial
- Solve a polynomial in factored form by setting it equal to zero

In this section we will apply factoring a monomial from a polynomial to solving polynomial equations. Recall that not all of the techniques we use for solving linear equations will apply to solving polynomial equations, so we will be using the zero product principle to solve for a variable.

We will begin with an example where the polynomial is already equal to zero.

### Example

Solve:

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

Notice how we were careful with signs in the last example. Even though one of the terms was negative, we factored out the positive common term of t. In the next example we will see what to do when the polynomial you are working with is not set equal to zero. Int eh followign video, we present more examples of solving quadratic equations by factoring.

### Example

Solve: [latex]6t=3t^2-12t[/latex]

The video that follows provides another example of solving a polynomial equation using the zero product principle and factoring.

We will work through one more example that is similar to the ones above, except this example has fractions and the greatest common monomial is negative.

### Example

Solve [latex]\frac{1}{2}y=-4y-\frac{1}{2}y^2[/latex]

Wow! In the last example, we used *many* skills to solve one equation. Let’s summarize them:

- We needed a common denominator to combine the like terms [latex]-4y\text{ and }-\frac{1}{2}y[/latex], after we moved all the terms to one side of the equation
- We found the GCF of the terms [latex]-\frac{9}{2}y\text{ and }-\frac{1}{2}y^2[/latex]
- We used the GCF to factor the polynomial [latex]-\frac{9}{2}y-\frac{1}{2}y^2[/latex]
- We used the zero product principle to solve the polynomial equation [latex]0=-\frac{1}{2}y\left(9+y\right)[/latex]

Sometimes solving an equation requires the combination of many algebraic principles and techniques. The last facet of solving the polynomial equation [latex]\frac{1}{2}y=-4y-\frac{1}{2}y^2[/latex] that we should talk about is negative signs.

We found that the GCF [latex]-\frac{1}{2}y[/latex] contained a negative coefficient. This meant that when we factored it out of all the terms in the polynomial, we were left with two positive factors, 9 and y. This explains why we were left with [latex]\left(9+y\right)[/latex] as one of the factors of our final product.

In the following video we present another example of solving a quadratic polynomial with fractional coefficients using factoring and the zero product principle.

In the next unit, we will learn more factoring techniques that will allow you to be able to solve a wider variety of polynomial equations such as [latex]3x^2-x=2[/latex].

## Summary

In this section we practiced using the zero product principle as a method for solving polynomial equations. We found that the techniques we used to solve linear equations did not give us the correct answer when used to solve a polynomial equation. We also found that a polynomial can be rewritten as a product by factoring out the greatest common factor. We used both factoring and the zero product principle to solve second degree polynomials.