Factoring Special Products

Goal 1 Recognize Special Products

Goal 2 Factor special products using patterns.

10.7

Recall what happens when you multiply the following:

(x + 4)(x + 4)

(x – 3)2

The results are called ____________________________.

Factoring a Perfect Square Trinomial                        

OR                        

 

It has to be exactly in this form to use this rule.  When you have a base being squared plus or minus twice the product of the two bases plus another base squared, it factors as the sum (or difference) of the bases being squared. 

Example 1 Factor the perfect square trinomial:

If you can recognize that it fits the form of a  perfect square trinomial, you can save yourself some time.

 

                      

 

*Fits the form of a perfect square trinomial *Factor as the sum of bases squared

Example 2 Factor the following trinomials:

x2 + 5x + 12 x2 + 6x + 9

Example 3 Factor the following trinomials:

x2 + 8x + 16 4x2 + 12x + 9

Factor the trinomial:Example 4

                           

Factoring a Difference of Two Squares                        

 

Just like the perfect square trinomial, the difference of two squares  has to be exactly in this form to use this rule.

Note that the sum of two squares DOES NOT factor.

Recall the product: (x + 2)(x – 2)=

Example 5 Factor the difference of two squares:

First note that there is no GCF to factor out of this polynomial.

This fits the form of a the difference of two squares. 

             

Factor the difference of two squares:Example 6

               

9x2 – 25

Factor the difference of two squares: (don’t forget the GCF if there is one.)

Example 6

               

8x2 – 32 9x2 – 36

Factoring a Sum of Two Cubes

                              

     

The sum of two cubes  has to be exactly in this form to use this rule.

1. When you have the sum of two cubes, you have a product of a binomial and a trinomial.

2. The binomial is the sum of the bases that are being cubed.

3. 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. 

          

Factor the sum of cubes:Example 7

First note that there is no GCF to factor out of this polynomial.

This fits the form of  the sum of cubes. 

                      

          

Factor the sum of cubes:(GCF first, if needed)

Example 8

                      

5x3 + 40 27x3 + 8

Factoring a Difference of Two Cubes

                              

       

The difference of two cubes has to be exactly in this form to use this rule.

1. When you have the difference of two cubes, you have a product of a binomial and a trinomial.

2. The binomial is the difference of the bases that are being cubed.

3. 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.

           

Factor the difference of cubes:Example 9

First note that there is no GCF to factor out of this polynomial.

This fits the form of  the difference of cubes.

                      

Example 10 Factor

8x3 – y3 54x3 – 81y3

Factoring Strategy

I.  GCF: Always check for the GCF first, no matter what.

II.  Binomials:

III. Trinomials:

a.             

b. Trial and error: c.  Perfect square trinomial:

IV.  Polynomials with four terms: factor by grouping.