CH 62 FACTORING QUADRATICS AN INTRODUCTION

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CH 62 FACTORING QUADRATICS,AN INTRODUCTION I NTRODUCTIONWe’re now ready to undertake the task of factoring additionalquadratic expressions. This skill will not only enable us to solvemany quadratic equations without resorting to the Quadratic Formula,but will allow us to reduce more kinds of algebraic fractions.To set the stage for factoring binomials (two terms) and trinomials(three terms), we’ll do one more example of double distributing toanalyze its inner workings:(2x 3)(x 5)i)Multiply the FIRST terms in each set of parentheses: 2xand x. The product is 2x 2 .ii)Multiply the OUTER terms: 2x and 5. The product is 10x.iii)Multiply the INNER terms: 3 and x. The product is 3x.iv)Multiply the LAST terms in each set of parentheses: 3 and5. The product is 15.v)Writing out these four products, followed by combining liketerms, gives us2x 2 10x 3x 15FirstOuterInnerLast 2x 2 13x 15

2To reiterate, we have(2x 3)(x 5) 2x 2 product ofFirst terms 10x 3x 15product of product ofproduct ofOuter terms Inner terms Last terms2x 2 13x 15The key idea to absorb here is that the 2x 2 in the answer is theproduct of the first terms, while the 15 is the product of the lastterms. And even more importantly, the middle term in theanswer, 13x, is the sum of the outer and inner products.Homework1.2.Find the following products -- do all the work in your head:a. (x 2)(x 3)b. (n 3)(n 5)c. (a 7)(a 5)d. (b 3)(b 10)e. (y 6)(y 6)f. (z 3)(z 3)g. (u 1)(u 7)h. (v 7)(v 7)i. (x 8)(x 8)j. (k 10)(k 10)Find the following products -- do all the work in your head:a. (2x 1)(x 4)b. (3n 3)(2n 5)c. (a 9)(3a 1)d. (6y 1)(2y 7)e. (2m 7)(2m 7)f. (4w 5)(4w 5)g. (5x 1)(5x 1)h. (2n 3)(7n 10)i. (7u 3)(7u 3)j. (12a 13)(12a 13)Ch 62 Factoring Quadratics, an Introduction

3 U NSCRAMBLING THE E GGNow suppose that we’ve been handed the quadratic expression2x 2 13x 15to factor. We’ve done enough double distributing to realize that thefactorization probably looks like this:(slot 1 slot 2) (slot 3 slot 4)where the contents of the four slots have to be determined.We know that the product of slot 1 and slot 3 must be 2x 2 . It’s logicalto assume that slot 1 is 2x and slot 3 is x. Now we have(2x slot 2) (x slot 4)We also know that the product of slot 2 and slot 4 must be 15. Nowwe’re at a crossroads: The product of 3 and 5 is 15, but the product of 1and 15 is also 15. It gets worse -- even when we select a pair of factorsof 15 to try, the order in which we place those factors in the slots mightbe crucial. As we try various combinations of numbers, keep in mindthat we’re trying to factor 2x 2 13x 15.Let’s try 1 and 15 (in that order):(2x 1)(x 15) 2x 2 30x x 15 2x 2 31x 15 Flip around the 1 and the 15:(2x 15)(x 1) 2x 2 2x 15x 15 2x 2 17x 15 That rules out using 1 and 15. Let’s use the 5 and the 3 (in that order):(2x 5)(x 3) 2x 2 6x 5x 15 2x 2 11x 15 Flip the 5 and 3:(2x 3)(x 5) 2x 2 10x 3x 15 2x 2 13x 15 And so the factorization of 2x 2 13x 15 is (2x 3)(x 5).Ch 62 Factoring Quadratics, an Introduction

4Recall that we can summarize the process we just completed in two ways: Inchanging2x 2 13x 15 to (2x 3)(x 5)we have1)converted an expression whose final operation is addition into anexpression whose final operation is multiplication, and we have2)converted an expression with three terms into an expression with oneterm.Either way, we’ve accomplished the goal of factoring.Homework3.True/False:a.x 2 5x 6 factors into (x 3)(x 2).b.y 2 16 factors into (y 4)(y 4).c.x 2 9x 15 factors into (x 3)(x 5).d.n 2 6n 9 factors into (n 3)(n 3).e.2a 2 11a 6 factors

Ch 62 Factoring Quadratics, an Introduction To reiterate, we have (2x 3)(x 5) 2x 2 10x 3x 15 product of product of product of product of First terms Outer terms Inner terms Last terms 2x 2 13x 15 The key idea to absorb here is that the 2x 2 in the answer is the

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