Last time, we learned how to find the greatest common factor of two monomials. Let’s take a quick review.

You are given two monomials. Find their greatest common factor.

After selecting your answer, click the “Submit” button.

Now, let’s continue with today’s lesson.

Here we have a polynomial with three monomial terms: <EQUATION>

Next, we show the prime factorization of each term.

What is the greatest common factor of the monomial terms? Click the “Submit” button after selecting your answer.

The greatest common factor is 2xy.

Next, we write each term as the product of the greatest common factor and another factor. This gives <EQUATION><EQUATION>.

Finally, we use the distributive property. The polynomial is expressed as the product of the greatest common factor and the remaining polynomial: <EQUATION>

A polynomial that is written as the product of monomials and polynomials is said to be in factored form.

Now, you try some examples.

You are given a polynomial. Use the distribution property to find its factored form.

After selecting your answer, click the “Submit” button.

Here’s a polynomial with four terms: 12xz + 8z + 15xy + 10y. Let’s factor it using a different method.

First, we group together pairs of terms that have common factors.

Next, we find the greatest common factor for each pair. For the first pair, it is 4z. For the second pair, it is 5y.

Then, we factor each pair using its greatest common factor.

The polynomial now has two terms.

These terms have the common factor 3x + 2.

We can factor this polynomial using the distributive property.

The result is the product of two binomials: (4z + 5y)(3x + 2).

This is an example of factoring by grouping.

We can use this method to factor some polynomials with four or more terms.

Let’s look at a problem to solve by factoring. Maria’s house has a large, rectangular picture window.

The area of the window is represented by a polynomial with four terms: <EQUATION>

Here are two binomials representing the length and width of the window. Find the missing values by factoring the area polynomial. Click the “Submit” button after entering the values into the text boxes.

The length of the window is 3x + 2y, and the width is x + 3. We can check the these by using the FOIL method.

The product gives the original polynomial for the area.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education