Learning Objectives
Introduction
In this tutorial we help you expand your knowledge of polynomials by looking at multiplying polynomials together. We will look at using the distributive property, initially shown in Tutorial 5: Properties of Real Numbers, to help us out. Again, we are using a concept that you have already seen to apply to the new concept. After going through this tutorial you should have multiplying polynomials down pat.
Tutorial
OR
This is the reverse of the binomial squared found in Tutorial 26: Multiplying Polynomials. Recall that factoring is the reverse of multiplication.
Since it is a trinomial, you can try factoring this by trial and error as shown in Tutorial 28: Factoring Trinomials. But if you can recognize that it fits the form of a perfect square trinomial, you can save yourself some time.
Since it is a trinomial, you can try factoring this by trial and error as shown in Tutorial 28 (Factoring Trinomials). But if you can recognize that it fits the form of a perfect square trinomial, you can save yourself some time.
Just like the perfect square trinomial, the difference of two squares has to be exactly in this form to use this rule. When you have the difference of two bases being squared, it factors as the product of the sum and difference of the bases that are being squared.
This is the reverse of the product of the sum and difference of two terms found in Tutorial 26: Multiplying Polynomials. Recall that factoring is the reverse of multiplication.
This fits the form of a the difference of two squares. So we will factor using that rule:
This fits the form of the difference of two squares. So we will factor using that rule:
This fits the form of the sum of cubes. So we will factor using that rule:
The difference of two cubes has to be exactly in this form to use this rule. When you have the difference of two cubes, you have a product of a binomial and a trinomial. The binomial is the difference of the bases that are being cubed. The trinomial is the first base squared, the second term is the opposite of the product of the two bases found, and the third term is the second base squared.
This fits the form of the difference of cubes. So we will factor using that rule:
When you need to factor, you ALWAYS look for the GCF first. Whether you have a GCF or not, then you continue looking to see if you have anything else that factors.
Below is a checklist to make sure you do not miss anything. Always factor until you can not factor any further.
Factoring StrategyI. GCF:
II. Binomials:
b.
c.
III. Trinomials:
b. Trial and error:
c. Perfect square trinomial:
IV. Polynomials with four terms:
The GCF. In this case, there is one.
Factoring out the GCF of 4 as was shown in Tutorial 27: The GCF and Factoring by Grouping, we get:
Note that if we would multiply this out, we would get the original polynomial.
The GCF. In this case, there is not one.
So we assess what we have. It fits the form of a difference of two squares, so we will factor it accordingly:
Note that if we would multiply this out, we would get the original polynomial.
The GCF. In this case, there is not one.
So we assess what we have. It fits the form of a sum of two cubes, so we will factor it accordingly:
Note that if we would multiply this out, we would get the original polynomial.
The GCF. In this case, there is not one.
So we assess what we have. This is a trinomial that does not fit the form of a perfect square trinomial. Looks like we will have to use trial and error as shown in Tutorial 28: Factoring Trinomials:
Note that if we would multiply this out, we would get the original polynomial.
The GCF. In this case, there is not one.
So we assess what we have. This is a polynomial with four terms. Looks like we will have to try factoring it by grouping as shown in Tutorial 27: The Greatest Common Factor and Factoring by Grouping:
Note that if we would multiply this out, we would get the original polynomial.
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1e: Factor Completely.
Need Extra Help on these Topics?
http://www.sosmath.com/algebra/factor/fac05/fac05.html
This webpage helps you with the factoring by special products discussed
in this tutorial.
http://www.purplemath.com/modules/specfact.htm
This webpage helps you with the factoring by special products discussed
in this tutorial.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on July 15, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.