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.


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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 here

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