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§ 6.3

§ 6.3. Special Products. Special Products. In this section we will use the distributive property to develop patterns that can help you in multiplying some special binomials quickly.

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§ 6.3

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  1. §6.3 Special Products

  2. Special Products In this section we will use the distributive property to develop patterns that can help you in multiplying some special binomials quickly. We will find the product of two binomials using a method called FOIL. You will be thinking…”first two, outer two, inner two, last two” before the section is over. We will learn a formula for finding the square of a binomial sum. You will learn a formula for finding the product of the sum and difference of two terms. And whether you choose to take a handy shortcut and use these formulas or simply use polynomial multiplication will be left to you to decide. Blitzer, Introductory Algebra, 5e – Slide #2 Section 6.3

  3. Multiplying Polynomials - FOIL last F O I L first inside Product of First terms Product of Outside terms Product of Inside terms Product of Last terms outside Blitzer, Introductory Algebra, 5e – Slide #3 Section 6.3

  4. The FOIL Method Product of Two Binomials Distribute each term in the first binomial through each term of the second binomial. (ax + b)(cx + d) = ax·cx + ax·d + b·cx + b·d Product of the first terms Product of the outside terms Product of the inside terms Product of the last terms Blitzer, Introductory Algebra, 5e – Slide #4 Section 6.3

  5. Multiplying Polynomials - FOIL EXAMPLE Multiply SOLUTION last F O I L first Multiply inside outside Combine like terms Blitzer, Introductory Algebra, 5e – Slide #5 Section 6.3

  6. FOIL Method Multiply (5x + 2)(x + 7) EXAMPLE (5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7 =5x2 + 35x + 2x +14 =5x2 + 37x +14 F O I L Product of the last terms Product of the first terms Product of the outside terms Product of the inside terms Blitzer, Introductory Algebra, 5e – Slide #6 Section 6.3

  7. FOIL Method Multiply (3x - 5)(2x - 8) EXAMPLE (3x - 5)(2x - 8) = 3x·2x + 3x·(-8) + (-5)·2x + (-5)·(-8) =5x2 + 35x + 2x +14 =5x2 + 37x +14 First two, Outer two, Inner two, Last two… F O I L Blitzer, Introductory Algebra, 5e – Slide #7 Section 6.3

  8. The Square of a Binomial Sum (A + B)2 = A2 + 2AB + B2 The square of a binomial sum is the first term squared plus two times the product of the terms plus the last term squared. Blitzer, Introductory Algebra, 5e – Slide #8 Section 6.3

  9. Multiplying the Sum and Difference of Two Terms (A + B)(A – B) = A2 - B2 The product of the sum and the difference of the same two terms is the square of the first term minus the square of the second term. Blitzer, Introductory Algebra, 5e – Slide #9 Section 6.3

  10. Product of the Sum and Difference of Two Terms Multiply (x – 5)(x + 5) Since this is the product of a sum and a difference, we use the rule: (A + B)(A – B) = A2 - B2 (x – 5)(x + 5) = x2 - 52 = x2 - 25 EXAMPLE Blitzer, Introductory Algebra, 5e – Slide #10 Section 6.3

  11. The Square of a Binomial Sum Find (x + 7 )2 EXAMPLE Since this is the square of a binomial sum, we use the rule: (A + B)2 = A2 + 2AB + B2 (x + 7)2 = x2 + 2x(7) + 72 = x2 + 14x + 49 Blitzer, Introductory Algebra, 5e – Slide #11 Section 6.3

  12. The Square of a Binomial Difference (A - B)2 = A2 - 2AB + B2 The square of a binomial difference is the first term squared minus two times the product of the terms plus the last term squared. Blitzer, Introductory Algebra, 5e – Slide #12 Section 6.3

  13. The Square of a Binomial Difference Find (3x - 5 )2 Since this is the square of a binomial difference, we use the rule: (A - B)2 = A2 - 2AB + B2 (3x - 5)2 = (3x)2 - 2·3x(5) + 52 = 9x2 - 30x + 25 EXAMPLE Blitzer, Introductory Algebra, 5e – Slide #13 Section 6.3

  14. The Square of a Binomial Sum Find (x + 7 )2 EXAMPLE Since this is the square of a binomial sum, we use the rule: (A + B)2 = A2 + 2AB + B2 (x + 7)2 = x2 + 2x(7) + 72 = x2 + 14x + 49 Blitzer, Introductory Algebra, 5e – Slide #14 Section 6.3

  15. The Square of a Binomial Difference Find (3x - 5 )2 Since this is the square of a binomial difference, we use the rule: (A - B)2 = A2 - 2AB + B2 (3x - 5)2 = (3x)2 - 2·3x(5) + 52 = 9x2 - 30x + 25 EXAMPLE Blitzer, Introductory Algebra, 5e – Slide #15 Section 6.3

  16. Multiplying Polynomials – Special Formulas Blitzer, Introductory Algebra, 5e – Slide #16 Section 6.3

  17. Multiplying Polynomials – Special Formulas EXAMPLE Multiply SOLUTION Use the special-product formula shown. Blitzer, Introductory Algebra, 5e – Slide #17 Section 6.3

  18. Multiplying Polynomials – Special Formulas EXAMPLE Multiply SOLUTION Use the special-product formula shown. Blitzer, Introductory Algebra, 5e – Slide #18 Section 6.3

  19. Multiplying Polynomials – Special Formulas EXAMPLE Multiply SOLUTION Use the special-product formula shown. - = First Term Squared Second Term Squared Product = Blitzer, Introductory Algebra, 5e – Slide #19 Section 6.3

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