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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 12 Exponents and Polynomials

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 12.6 Special Products

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Martin-Gay, Developmental Mathematics, 2e 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The FOIL Method When multiplying 2 binomials, the distributive property can be easily remembered as the FOIL method. F – product of First terms O – product of Outside terms I – product of Inside terms L – product of Last terms

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Martin-Gay, Developmental Mathematics, 2e 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. = y 2 – 8y – 48 Multiply (y – 12)(y + 4). (y – 12)(y + 4) Product of First terms is y 2 Product of Outside terms is 4y Product of Inside terms is – 12y Product of Last terms is – 48 (y – 12)(y + 4) = y 2 + 4y – 12y – 48 F O I L Example

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Martin-Gay, Developmental Mathematics, 2e 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply (2x – 4)(7x + 5). (2x – 4)(7x + 5) = = 14x 2 + 10x – 28x – 20 F 2x(7x) F + 2x(5) O – 4(7x) I – 4(5) L O I L = 14x 2 – 18x – 20 We multiplied these same two binomials together in the previous section, using a different technique, but arrived at the same product. Example

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Martin-Gay, Developmental Mathematics, 2e 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. In the process of using the FOIL method on products of certain types of binomials, we see specific patterns that lead to special products. Special Products

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Martin-Gay, Developmental Mathematics, 2e 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Squaring a Binomial A binomial squared is equal to the square of the first term plus or minus twice the product of both terms plus the square of the second term. (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 – 2ab + b 2 Special Products

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Martin-Gay, Developmental Mathematics, 2e 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Example Multiply. (x + 6) 2 F OI L (x + 6) 2 = x 2 + 6x + 6x + 36 = x 2 + 12x + 36 The inner and outer products are the same. = (x + 6)(x + 6)

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Martin-Gay, Developmental Mathematics, 2e 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Example a. (12a – 3) 2 = 144a 2 – 72a + 9 = (12a) 2 – 2(12a)(3) + (3) 2 b. (x + y) 2 = x 2 + 2xy + y 2 Multiply.

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Martin-Gay, Developmental Mathematics, 2e 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiplying the Sum and Difference of Two Terms The product of the sum and difference of two terms is the square of the first term minus the square of the second term. (a + b)(a – b) = a 2 – b 2 Special Products

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Martin-Gay, Developmental Mathematics, 2e 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Example Multiply. = (2x)(2x) + (2x)(– 4) + (4)(2x) + (4)(– 4) F OI L (2x + 4)(2x – 4) = 4x 2 + (– 8x) + 8x + (–16) = 4x 2 – 16 The inner and outer products cancel.

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Martin-Gay, Developmental Mathematics, 2e 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Example Multiply. a. (5a + 3)(5a – 3) = 25a 2 – 9 = (5a) 2 – 3 2 b. (8c + 2d)(8c – 2d) = 64c 2 – 4d 2 = (8c) 2 – (2d) 2

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Martin-Gay, Developmental Mathematics, 2e 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Helpful Hint When multiplying two binomials, you may always use the FOIL order or method. When multiplying any two polynomials, you may always use the distributive property to find the product.

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