Factoring Polynomials
Factoring Binomials
You are familiar with using a factor tree to factor integers. Now build upon that knowledge as you discover how to factor binomial expressions. You will use the distributive property to represent a binomial as the product of the greatest common factor of its terms and a remaining factor. Click the player button to begin.
View a printable version of this interactivity.
Factoring Trinomials With a Leading Coefficient of One
Get ready to expand your knowledge of factoring as you learn how to factor second-degree trinomials that have a leading coefficient equal to one. Your skills performing integer operations and your knowledge of the distributive property will be useful tools. Click the player button to begin.
View a printable version of this interactivity.
Factoring Trinomials With a Greatest Common Factor
You have learned how to factor a trinomial in the form ax2 + bx + c, where a = 1. When the leading coefficient is not equal to 1, there are different strategies available to help you factor. Begin by determining if the terms of the trinomials have a greatest common factor. If they do, factor it out of the expression, and factor the trinomial completely. Take a moment to look at the example.
| 2x2 + 10x + 12 | Identify the greatest common factor of the terms of the trinomial. |
| 2(x2 + 5x + 6) | Factor out the greatest common factor: 2. |
| 2(x + 2)(x + 3) | Factor the trinomial completely. |
Factoring Trinomials With a Leading Coefficient Not Equal to One
Factoring by grouping is a technique that is used to factor trinomials with a leading coefficient not equal to one. Prepare to explore this technique as you stretch your factoring skills. Click the player button to begin.
View a printable version of this interactivity.
Using Factoring to Determine Quotients
Take a moment to familiarize yourself with the example below and see how factoring can help you simplify quotients.
Example
Simplify:
x2− 11x + 30x − 6 |
Factor the trinomial in the numerator completely.
x2− 11x + 30x − 6 |
= | (x − 5)(x − 6)(x − 6) |
Cancel out the binomial that is common to the numerator and denominator.
Using Graphs to Verify Factors of a Polynomial
Functions can be defined by polynomial expressions that in turn can be represented graphically. The graphing calculator can be used as a tool to factor and verify the factors of these polynomials. Are you ready to extend your skills using the graphing calculator to analyze graphs? Click the player button to begin.
View a printable version of this interactivity.
x-Intercepts and Factors
Consider the function defined in the Self-Check of the previous interactivity, y = x2 + 4x − 5, represented by the graph below.
In the interactivity, you used the graphing calculator to verify the factors (x + 5) and (x − 1).
You can also determine the factors of a polynomial function graphically. If a is an x-intercept, then (x − a) is a factor.
- 1 is an x-intercept, therefore (x − 1) is a factor.
- −5 is an x-intercept, therefore (x + 5) is a factor.
A Note About Prime Polynomials
There are some polynomials that cannot be factored using the techniques that you explored in this topic. These polynomials are known as prime polynomials. They cannot be represented as the product of two or more polynomials that are of a lesser degree than the original polynomial, and whose coefficients and terms consist only of integers. For example, the polynomial x2 + 2x + 4 is a prime polynomial. Can you think of another example of a prime polynomial?
Factoring Polynomials Review
Now that you have explored factoring polynomials, review your knowledge in this interactivity. Click the player button to get started.
Did you answer the content review questions incorrectly? Do you want more instruction or extra practice? If so, view the videos Factoring Binomials, Factoring Trinomials with a Leading Coefficient of 1, Factoring Trinomials with a Leading Coefficient Not Equal to 1, and Using the Graphing Calculator to Verify Factors of Polynomials from eMediaVASM.