Section P.6Factoring Trinomials61Factoring Trinomials of the Form Try covering the factored forms in the left-hand column below. Can you deter-mine the factored forms from the trinomial forms?Factored FormFOILTrinomial FormYour goal here is to factor trinomials of the form To begin, considerthe factorizationBy multiplying the right-hand side, you obtain the following result.Sum ofProduct oftermstermsSo, to factora trinomial into a product of two binomials, you mustfind two factors of cwith a sum of b.Example 1Factoring TrinomialsFactor the trinomials (a)and (b)Solution(a)You need to find two factors whose product is and whose sum is The product of and 2 is The sum of and 2 is (b)You need to find two factors whose product is 6 and whose sum is The product of and is 6The sum of and is Now try Exercise 7.Note that when the constant term of the trinomial is positive, its factors musthave likesigns; otherwise, its factors have unlikesigns.5.23x25x6x3x2235.2.4x22x8x4x28.42.8x25x6.x22x8x2bxcbxcx2x2mn xmnxmxnx2nxmxmnx2bxcxmxn.x2bxc.3x5x13x23x5x53x28x5x3x2x22x3x6x25x6x1x4x24xx4x23x4x2bxcFactoring TrinomialsP.6•Factor trinomials of the form•Factor trinomials of the form•Factor trinomials by grouping•Factor perfect square trinomials•Select the best factoring technique using the guidelinesfor factoring polynomialsThe techniques for factoring trinomials will help you in solvingquadratic equations.ax2bxcx2bxcWhatyou should learn:Whyyou should learn it:Study TipUse a list to help you find thetwo numbers with the requiredproduct and sum. For Example1(a):Factors of Sum1,872,42Because is the requiredsum, the correct factorization is x22x8x4x2 .22,241,788

62Chapter PPrerequisitesWhen factoring a trinomial of the form if you have troublefinding two factors of cwith a sum of b, it may be helpful to list all of the distinctpairs of factors and then choose the appropriate pair from the list. For instance,consider the trinomialFor this trinomial,and So, you need to find two factors ofwith a sum of as shown at the left. With experience, you will be able to narrow this list down mentallyto only two or three possibilities whosesums can then be tested to determine the correct factorization, which isExample 2Factoring a TrinomialFactor the trinomial SolutionTo factor this trinomial, you need to find two factors whose product is andwhose sum is The product of 2 and is The sum of 2 and is Now try Exercise 11.Applications of algebra sometimes involve trinomials that have a commonmonomial factor. To factor such trinomials completely, first factor out thecommon monomial factor. Then try to factor the resulting trinomial by themethods given in this section.