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Polynomials: Special ProductsExplanation & Practice

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483
-1- Polynomials: Special Products Explanation and Practice Example 1. Find the square of (3x 2). This problem asks us to find the square of a binomial. Solution: To square 3x 2, we multiply it by itself: (3x 2) 2 = (3x 2)(3x 2) Definition of exponents = 9x 2

6x 6x + 4 FOIL method

= 9x 2

12x + 4 Combine like terms

Notice that the first and last terms in the answer are the square of the first and last terms in the original problem and that the middle term is twice the product of the two terms in the original binomial.

Example 2. (a + b)

2 = (a + b)(a + b) = a2 + 2ab + b 2

Example 3. (a b)

2 = (a b)(a b) = a2 2ab + b 2 Binomial squares having the form of Examples 2 and 3 occur very frequently in algebra. It will be to your advantage to memorize the following rule for squaring a binomial: RULE: The square of a binomial is the sum of the square of the first term, the square of the last term, and twice the product of the two original terms. In symbols, this rule is written as follows: (x + y) 2 = x 2 + 2xy + y 2

Square Twice Square

of product of first of the last term two terms term

Examples: Multiply using the preceding rule.

First term Twice their Last term

squared product squared Answer

4. (x 5)

2 = x 2 + 2(x)(-5) + 25 = x 2

10x + 25

5. (x + 2)

2 = x 2 + 2(x)(2) + 4 = x 2 + 4x + 4

6. (2x 3)

2 = 4x 2 + 2(2x)(3) + 9 = 4x 2

12x + 9

7. (5x 4)

2 = 25x 2 + 2(5x)(4) + 16 = 25x 2

40x + 16

Your work will be faster and easier if you use the rule for this special product rather than taking the time to do FOIL. Polynomials: Special ProductsExplanation & Practice

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-2- Another special product that occurs frequently is (a + b)(a b). The only difference in the two binomials is the sign between the two terms. The interesting thing about this type of product is that if you use FOIL, the middle term is always zero. Here are some examples:

Examples: Multiply using the FOIL method

8. (2x 3)(2x + 3) = 4x

2 + 6x 6x 9 FOIL method = 4x 2 9

9. (x 5)(x + 5) = x

2 + 5x 5x 25 FOIL method = x 2 25

10. (3x 1)(3x + 1) = 9x

2 + 3x 3x 1 FOIL method = 9x 2 1

Notice that in each case the middle term is zero

The answers all turn out to be the difference of two squares. Here is a rule to help you memorize the result: RULE: When multiplying two binomials that differ only in the sign between their terms, subtract the square of the last term from the square of the first term. Or (a b)(a + b) = a 2 b 2 Here are some problems that result in the difference of two squares. Examples: Multiply using the preceding rule. Avoid using FOIL: save time!

11. (x + 3)(x 3) = x

2 9

12. (a + 2)(a 2) = a

2 4

13. (9a + 1)(9a 1) = 81a

2 1

14. (2x 5y)(2x + 5y) = 4x

2 25y
2

15. (3a 7b)(3a + 7b) = 9a

2 49b
2 Although all the problems in this section can be worked correctly using FOIL, they can be done much faster if the two rules are memorized. Here is a summary of the two rules that apply to our (a + b) 2 = (a + b)(a + b) = a 2 + 2ab + b 2 (a b) 2 = (a b)(a b) = a 2

2ab + b

2 (a b)(a + b) = a 2 b 2 Polynomials: Special ProductsExplanation & Practice

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-3- Example 16. Write an expression in symbols for the sum of the squares of three consecutive even integers. Then simplify that expression. Solution: If we let x = the first of the even integers, then x + 2 is the next consecutive even integer, and x + 4 is the one after that. An expression for the sum of their squares is x 2 + (x + 2) 2 + (x + 4) 2 = x 2 + (x 2 + 4x + 4) + (x 2 + 8x + 16) Expand squares = 3x 2 + 12x + 20 Add like terms

Practice

Perform the indicated operations using the rules for special products (not FOIL).

1. (x 2)2 2. (a + 3)2 3. (x 5)2 4. (x + 4)2

5. (a

2 1 )2 6. (x + 10)2 7. (x 10)2 8. (a + .8)2

9. (2x 1)2 10. (3x + 2)2 11. (3a + 5b)2 12. (5a 3b)2

13. (2m 7n)2 14. (x2 + 5)2 15. (a2 2)2 16. (x3 + 4)2

17. (x 3)(x + 3) 18. (x + 4)(x 4) 19. (a + 5)(a 5) 20. (a 6)(a + 6)

21. (y 1)(y + 1) 22. (y 2)(y + 2) 23. (9 + x)(9 x) 24. (10 x)(10 + x)

25. (2x + 5)(2x 5) 26. (3x + 5)(3x 5) 27. (4x +

3 1 )(4x 3 1

28. (6 7x)(6 + 7x) 29. (a2 + 4)(a2 4) 30. (5y4 8)(5y4+ 8)

For more practice, see Handout 484.

Polynomials: Special ProductsExplanation & Practice

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-4-

Answer Key

Polynomials: Special Products Explanation and Practice

1. x

2

4x + 4 2. a2 + 6a + 9 3. x

2

10x + 25

4. x2 + 8x + 16 5. a

2 a + 4 1

6. x

2 + 20x + 100 7. x 2

20x + 100 8. a

2 + 1.6a + .64 9. 4x 2

4x + 1

10. 9x

2 + 12x + 4 11. 9a2 + 30ab+ 25b2 12. 25a 2

30ab + 9b

2

13. 4m

2

28mn + 49n

2

14. x

4 + 10x 2 + 25 15. a 4 4a 2 + 4

16. x

6 + 8x 3 + 16 17. x 2

9 18. x

2 16

19. a

2

25 20. a

2

36 21. y

2 1

22. y

2

4 23. 81 x

2

24. 100 x

2

25. 4x

2

25 26. 9x

2

25 27. 16x

2 9 1

28. 36 49x

2

29. a

4

16 30. 25y8 64

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