L> Factoring
 GCF and Factoring Trinomials Factoring Out ns Greatest Common Factor (GCF)Consider the example x (x - 3)= x2 - 3xWe can do the process in reverse:Look at x2 - 3x and notice that both have a common factor ofx.We can pull out the x terms (use the distributive property in reverse). x2 - 3x= x (x - 3).Example: Consider the expression 2x3 - 4x2 Find the GCF Pull it out. Solution: The GCF ins 2x2 We can write 2x3 - 4x2 =2x2(x - 2) Exercises: Find the GCF Pull it out. x2 - 5x 5xy + 25y 3x(2 - x) + 4(2 - x) 6x2y3 - 8x2y2 +10x4y4 Factoring through GroupingConsider the expression 3x2 + 6x - 4x - 8We can factor this by grouping two at a time: (3x2 + 6x) - (4x + 8)We now pull out the GCF of each: 3x (x + 2) -4 (x + 2)Pull out the GCF again: (3x -4) (x + 2)ExercisesFactor the following by grouping: 3x2 - 6x + x - 2 x3 - 3x2 + x - 3 Factoring Trinomials With constant Leading CoefficientConsider the trinomial: x2 + 5x + 4We want to factor this trinomial, that is do FOIL in reverse. Sincethe leading coefficient is 1 we can write: x2 + 5x + 4 = (x +a) (x + b) Wbelow a and also b are unknown numbers.FOIL the right expression out to get: x2 + 5x + 4 =x2 + bx + ax + ab =x2 + (a+ b) x + abHence we are searching for two numbers such that their product ins 4 and theirsum is 5. Note that the only pairs of number whose product is 4 are (2,2) and (1,4).Of these two pairs only (1,4) add up to 5.Hence x2 + 5x + 4= (x + 1) (x + 4) Two factor a trinomial with leading coefficient 1 we ask ourselves the followingquestions:What two numbers multiply to the last coefficient and add to the middlecoefficient?Example: Factor x2 - 3x - 10The pairs that multiply to -10 are (1,-10) and (-1,10)and (5,-2) and(2,-5)Only the last pair adds come -3 hence x2 - 3x - 10= (x + 2) (x - 5)Exercises: Factor if possible x2 + 8x + 15 x2 + 6x + 8 x2 - 4x - 5 x2 + 3x - 18 x2 - 7x +12 x2 - 4x - 9 Tthe AC MethodWhat can we do when the leading coefficient is not 1?We use an extension of factoring by grouping called ns ACmethod.Step by Step method for factoring Ax2 + Bx + C Step 1. Multiply together AC and list the factors of AC. Step 2. Find a pair that adds toB. If you cannot findsuch a pair then the trinomial does not factor. Step 3. Rewrite the middle term as a sum of terms whose coefficientsare the chosen pair. Step 4. Factor by grouping.You are watching: Gcf of 5x^2 20xSee more: What Does Green Porch Light Mean, The Story Behind The Green Porch Lights Remember you should always first pull out the GCFExample: Factor 2x2 + 5x - 25 AC = (2)(-25) = -50 the pairs are (1,-50) (-1,50) (2,-25) (-2,25) (5,-10) and (-5,10). We see that -5 + 10 =5 hence we choose the pair (-5,10) We create 2x2 - 5x + 10x - 25 (2x2 - 5x) + (10x - 25) = x (2x - 5) + 5 (2x - 5) = (x+ 5) (2x - 5) Exercises: Factor the following 15x2 - x -6 7x2 - 20x +12 30x3 + 25x2 + 5x 4x4 + 8x2 + 3 For an interactive applet on the AC method click here For an alternate method for the AC method click hereBackto the Factoring and Rational Expressions PageBack to the Basic Algebra Part II PageBack to the MathDepartment Home Pagee-mailQuestions and Suggestions