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Slide 1Slide 2Slide 3Product of Two BinomialsSlide 5Multiply (5x + 2)(x + 7)Multiply (3x - 5)(2x - 8)Slide 8Slide 9Multiply (x – 5)(x + 5)Find (x + 7 )2Slide 12Find (3x - 5 )2Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19§ 6.3Special ProductsSpecial ProductsBlitzer, Introductory Algebra, 5e – Slide #2 Section 6.3In 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 #3 Section 6.3Multiplying Polynomials - FOIL dbcxbdaxcxaxdcxbax Using the FOIL Method to Multiply BinomialsfirstoutsideinsidelastF O I LProduct of First termsProduct of Outside termsProduct of Inside termsProduct of Last termsProduct 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·dProduct of the first termsProduct of the outside termsProduct of the inside termsProduct of the last termsThe FOIL MethodBlitzer, Introductory Algebra, 5e – Slide #4 Section 6.3Blitzer, Introductory Algebra, 5e – Slide #5 Section 6.3Multiplying Polynomials - FOILEXAMPLEEXAMPLESOLUTIONSOLUTIONMultiply . 1534 xx 1534 xx13531454 xxxxCombine like termsMultiply3154202 xxx319202 xxF O I LfirstoutsideinsidelastMultiply (5x + 2)(x + 7)(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7=5x2 + 35x + 2x +14=5x2 + 37x +14Product of the first termsProduct of the outside termsProduct of the inside termsProduct of the last termsF O I LFOIL MethodBlitzer, Introductory Algebra, 5e – Slide #6 Section 6.3EXAMPLEXAMPLEEMultiply (3x - 5)(2x - 8)(3x - 5)(2x - 8) = 3x·2x + 3x·(-8) + (-5)·2x + (-5)·(-8)=5x2 + 35x + 2x +14=5x2 + 37x +14First two, Outer two, Inner two, Last two…F O I LFOIL MethodBlitzer, Introductory Algebra, 5e – Slide #7 Section 6.3EXAMPLEEXAMPLE(A + B)2 = A2 + 2AB + B2The square of a binomial sum is the first term squared plus two times the product of the terms plus the last term squared.The Square of a Binomial SumBlitzer, Introductory Algebra, 5e – Slide #8 Section 6.3(A + B)(A – B) = A2 - B2The 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.Multiplying the Sum and Difference of Two TermsBlitzer, Introductory Algebra, 5e – Slide #9 Section 6.3Multiply (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 Product of the Sum and Difference of Two TermsBlitzer, Introductory Algebra, 5e – Slide #10 Section 6.3EXAMPLEEXAMPLEFind (x + 7 )2Since 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 + 49The Square of a Binomial SumBlitzer, Introductory Algebra, 5e – Slide #11 Section 6.3EXAMPLEEXAMPLE(A - B)2 = A2 - 2AB + B2The square of a binomial difference is the first term squared minus two times the product of the terms plus the last term squared.The Square of a Binomial DifferenceBlitzer, Introductory Algebra, 5e – Slide #12 Section 6.3Find (3x - 5 )2Since 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 + 25The Square of a Binomial DifferenceBlitzer, Introductory Algebra, 5e – Slide #13 Section 6.3EXAMPLEEXAMPLEFind (x + 7 )2Since 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 + 49The Square of a Binomial SumBlitzer, Introductory Algebra, 5e – Slide #14 Section 6.3EXAMPLEEXAMPLEFind (3x - 5 )2Since 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 + 25The Square of a Binomial DifferenceBlitzer, Introductory Algebra, 5e – Slide #15 Section 6.3EXAMPLEEXAMPLEBlitzer, Introductory Algebra, 5e – Slide #16 Section 6.3Multiplying Polynomials – Special FormulasThe Square of a Binomial Sum 22BABABA The Square of a Binomial Difference 2222 BABABA 2222 BABABA The Product of the Sum and Difference of Two TermsBlitzer, Introductory Algebra, 5e – Slide #17 Section 6.3Multiplying Polynomials – Special FormulasEXAMPLEEXAMPLESOLUTIONSOLUTIONMultiply .42yx 24 yx 222 2 BABABA Use the special-product formula shown. + + = Product + +2TermFirstTerms theofProduct22TermLast 24xyx 422y22816 yxyx Blitzer, Introductory Algebra, 5e – Slide #18 Section 6.3Multiplying Polynomials – Special FormulasEXAMPLEEXAMPLESOLUTIONSOLUTIONMultiply . 432yx 243 yx 222 2 BABABA Use the special-product formula shown. - + = Product - +2TermFirstTerms theofProduct22TermLast 23xyx 432 24y2216249 yxyx Blitzer, Introductory Algebra, 5e – Slide #19 Section 6.3Multiplying Polynomials – Special FormulasEXAMPLEEXAMPLESOLUTIONSOLUTIONMultiply . 434322yxyyxy 2224 3 yxy 22 BABABA Use the special-product formula shown.First Term SquaredSecond Term SquaredProduct- ==242169 yyx