4.5 Factoring: Difference of Two Squares and Sum and Difference of Two Cubes BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 This pattern is referred to as the difference of two squares. Examples: Examples Term Perfect Square Note: The exponents of the variables or binomial terms are always even. In this lesson, our goal is to factor the binomial. Given the binomial, write it as the product of two factors. Objective A: Factoring the Difference of Two Squares Definition: A perfect square is the product of a term multiplied by itself.

4.5 Factoring: Difference of Two Squares and Sum and Difference of Two Cubes BobsMathClass.Com Copyright © 2010 All Rights Reserved. 2 A Square Root is one of two equal factors of a number. Number Square Root Reason Note that each term is a perfect square: 16 = 4  4, 4x 2 = (2x) 2, 121y 4 = (11y 2 ) 2, etc. The Difference of Two Squares The Difference of Squares is a binomial where both terms are perfect squares and the sign of the binomial is subtraction (hence the word “difference”).

4.5 Factoring: Difference of Two Squares and Sum and Difference of Two Cubes BobsMathClass.Com Copyright © 2010 All Rights Reserved. 3 Procedure: To factor the difference of two perfect squares: Step 1.Set up two sets of parentheses with one containing a “+” in the middle and the other a “–” in the middle. ( + )( – ) Step 2.Find the square root of the first perfect square and insert it in the first-term positions of the parentheses. Step 3.Find the square root of the second perfect square and insert it in the second- term positions of the parentheses. Step 4.Check by FOIL. Step 1. Write parentheses with +/– ( + )( – ) Step 2. Find square root of first perfect square. Insert in first-term positions. 4x Step 3. Find square root of second perfect square. Insert in second-term positions. 33 Step 4. Check by FOIL. (Not Shown) Your Turn Problem #1

4.5 Factoring: Difference of Two Squares and Sum and Difference of Two Cubes BobsMathClass.Com Copyright © 2010 All Rights Reserved. 4 Step 1. Write parentheses with +/– ( + )( – ) Step 2. Find square root of first perfect square. Insert in first-term positions. Step 3. Find square root of second perfect square. Insert in second-term positions. Step 4. Check by FOIL. (Not Shown) Your Turn Problem #2 Does this binomial follow the pattern of the difference of two squares? Answer: No. The formula is only for the difference of two squares. There is no formula for the sum of two squares. Your Turn Problem #3

4.5 Factoring: Difference of Two Squares and Sum and Difference of Two Cubes BobsMathClass.Com Copyright © 2010 All Rights Reserved. 5 Recall from the last section, the first step in any factoring problem is to first factor out the greatest common factor. After factoring out the greatest common factor, proceed to factor using the pattern for the difference of two squares. Once the pattern of the difference of two squares has been used, verify that the binomials in the answer are not another “difference of two squares”. If it is, use the same pattern when necessary to factor completely. Step 2-4. Factor difference of two squares Bring down common factor in front of parentheses. Step 1. Factor out GCF Step 5. Check by FOIL. (Not Shown) Your Turn Problem #4

4.5 Factoring: Difference of Two Squares and Sum and Difference of Two Cubes BobsMathClass.Com Copyright © 2010 All Rights Reserved. 6 Step 2. Factor difference of two squares Bring down common factor in front of parentheses. Step 1. Factor out GCF Step 3. Notice one of the binomials is the difference of two squares. Factor it and bring down other factors. Step 4. Multiply to check. Your Turn Problem #5

4.5 Factoring: Difference of Two Squares and Sum and Difference of Two Cubes BobsMathClass.Com Copyright © 2010 All Rights Reserved. 7 Objective B: Factoring the Sum and Difference of Two Cubes In the last objective, we covered a product of two binomials which resulted in the difference of two squares. In this objective, we want the product of two polynomials two obtain the sum or difference of two cubes. Example: which is the difference of two cubes. In the last objective, we had the pattern for the difference of two squares (there is no pattern for the sum of two squares). Here, we give the pattern for the sum and difference of two cubes. Sum and Difference of Two Cubes

4.5 Factoring: Difference of Two Squares and Sum and Difference of Two Cubes BobsMathClass.Com Copyright © 2010 All Rights Reserved. 8 A perfect cube is a term which can be written using as an expression raised to an exponent of three. Examples: Perfect Cubes Procedure: To factor the sum or difference of two cubes Step 1.Rewrite each term as a perfect cube. Usually, the expression will be written in parentheses raised to the third power on the outside of the parentheses. Step 2.Decide which formula will be used. If between the cubes is a “+”, use the sum of two cubes. If between the cubes is a “–”, use the difference of two cubes. Write down which formula will be used. Step 4.Simplify each polynomial. Step 3.Replace the “a” and “b” in the formula with the corresponding terms of the expression from Step 1. Step 5.Check by Multiplying

4.5 Factoring: Difference of Two Squares and Sum and Difference of Two Cubes BobsMathClass.Com Copyright © 2010 All Rights Reserved. 9 Step 2. Since the sign between the two cubes is a minus, we will use the formula for the difference of two cubes. Step 1. Rewrite as perfect cubes Step 4. Simplify Step 5. Multiply to check. (Not shown) Step 3. Replace “a” with “x” and “b” with “2” a=x b=2 Your Turn Problem #6

4.5 Factoring: Difference of Two Squares and Sum and Difference of Two Cubes BobsMathClass.Com Copyright © 2010 All Rights Reserved. 10 Step 2. Since the sign between the two cubes is a plus, we will use the formula for the sum of two cubes. Write the formula using “x” and “y” since the problem is already contains the variables “a” and “b”. Step 1. Rewrite as perfect cubes Step 4. Simplify Step 5. Multiply to check. (Not shown) Step 3. Replace “x” with “2a” and “y” with “3b” x=2a y=3b Your Turn Problem #7 The End. B.R