Polynomials: Factoring - XYZ Custom Plus

750Chapter 12 Polynomials: FactoringBFactoring Out the Greatest Common Factor3. Factor 2x 2 − 2x − 24.ExAmplE 3SOlutiOnThe coefficient of x 2 is 2. We begin by factoring out the greatest commonfactor, which is 2:Factor 2x 2 + 10x− 28.2x 2 + 10x− 28 = 2(x2 + 5x− 14)NoteIn Example 3 webegan by factoringout the greatestcommon factor. The first step infactoring any trinomial is to lookfor the greatest common factor.If the trinomial in question has agreatest common factor other than1, we factor it out first and then tryto factor the trinomial that remains.Now, we factor the remaining trinomial by finding a pair of numbers whose sumis 5 and whose product is −14. Here are the possibilities:From the last line we see that the factors of x 2 + 5x− 14 are (x+ 7) and (x− 2).Here is the complete solution:productsSums−1(14) = −14 −1 + 14 = 131(−14) = −14 1 + (−14) = −13−7(2) = −14 −7 + 2 = −57(−2) = −14 7 + (−2) = 52x 2 + 10x− 28 = 2(x2 + 5x− 14)= 2(x+ 7)(x− 2)4. Factor 3x3 + 18x2 + 15x.ExAmplE 4Factor 3x3 − 3x2 − 18x.SOlutiOn We begin by factoring out the greatest common factor, which is 3x.Then we factor the remaining trinomial. Without showing the table of productsand sums as we did in Examples 2 and 3, here is the complete solution:3x3 − 3x2 − 18x= 3x(x2 − x − 6)= 3x(x− 3)(x+ 2)5. Factor x 2 + 7xy+ 12y2 .NoteTrinomials in whichthe coefficient of thesecond-degree term is1 are the easiest to factor. Successin factoring any type of polynomialis directly related to the amountof time spent working the problems.The more we practice, themore accomplished we become atfactoring.Answers3. 2(x− 4)(x+ 3)4. 3x(x+ 5)(x+ 1)5. (x+ 3y)(x+ 4y)ExAmplE 5SOlutiOnFactor x 2 + 8xy+ 12y2 .This time we need two expressions whose product is 12y2 andwhose sum is 8y. The two expressions are 6yand 2y(see Example 1 in thissection):x 2 + 8xy+ 12y2 = (x+ 6y)(x+ 2y)You should convince yourself that these factors are correct by finding theirproduct.Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. When the leading coefficient of a trinomial is 1, what is the relationshipbetween the other two coefficients and the factors of the trinomial?2. When factoring polynomials, what should you look for first?3. How can you check to see that you have factored a trinomial correctly?4. Describe how you would find the factors of x 2 + 8x + 12.