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Factoring Trinomials THE X-BOX METHOD of the form ax2 + bx + c where a ≠ 1 and the GCF of a, b, c = 1 b=m+n GCF of ax2 and nx KFHS Algebra 2 GCF of mx and c b m n ac ac = m ∙ n GCF of ax2 and mx ax2 mx GCF of nx and c nx c Example: Factor 3x2 + 11x – 20 Step 1: Identify a, b, c and fill in b and ac in the X. a=3 Step 3: Draw your box and fill in ax2 , c, mx, nx. 15 + (-4) = 11 11 11 15 b = 11 c = -20 Step 2: Identify m & n such that their product is ac and their sum is b. In other words, m & n are factors of ac that add up to b. -60 11 15 -4 3x2 15x -4x -20 -4 -60 -60 15 ∙ (-4) = -60 Step 4: Find the GCF of each column and write it above the appropriate column. GCF of ax2 and nx, GCF of mx and c. x 5 3x2 15x -4x -20 Step 5: Find the GCF of each row and write it to the left of the appropriate row. GCF of ax2 and mx, GCF of nx and c. 3x -4 x 5 3x2 15x -4x Step 6: Write the factors from the top and left of the box in parentheses. x 5 3x 3x2 15x -4 -4x -20 (x + 5) (3x – 4) -20 Step 7: Multiply using FOIL to confirm the factors are correct. (x + 5) (3x – 4) 3x2 – 4x + 15x – 20 3x2 + 11x - 20 THE X-BOX METHOD KFHS Algebra 2 Factoring Trinomials The Slip & Slide Method of the form ax2 + bx + c where a ≠ 1 and the GCF of a, b, c = 1 1) Multiply c by a (SLIP the a to the end) 2) Rewrite problem with the new c and with a set to equal 1. 3) Find factors of the new c that add up to b. 4) Write the factors as a product. 5) SLIDE the original a back in (divide by a) 6) Reduce. If the fraction reduces evenly, do so. If the fraction does not reduce evenly, put the denominator in front of the x and leave the numerator where it is. 7) Use FOIL to check your factors; make sure that they multiply back to the original quadratic expression. Example: Factor 3x2 + 11x – 20 Step 1: Identify a, b, c and multiply a and c. SLIP the a to the end by multiplying. Step 2: Rewrite problem with new c and a set to 1 a = 3 b = 11 a*c = -60 x2 + 11x – 60 c = -20 Step 3: Find factors of the new c that add up to b The numbers that multiply to get -60 and add to get 11 are -4 and 15. On calculator, go to y= and put in -60/x. Hit 2nd Graph and look at the table to find the factors of -60. Find the two that add up to 11. Step 4: Write the factors as a product. (x – 4) (x + 15) Step 6: REDUCE If fractions reduce evenly, do so. If not, move denominator in front of the x, and leave the numerator. (𝟑𝒙 − 𝟒)(𝒙 + 𝟓) Step 5: SLIDE the 6 (the original value of a) back in by dividing. 𝟒 𝟏𝟓 (𝒙 − )(𝒙 + ) 𝟑 𝟑 Step 6: REDUCE 𝟒 (𝒙 − )(𝒙 + 𝟓) 𝟑 Step 7: Multiply using FOIL to confirm the factors are correct. (3x - 4) (x + 5) 3x2 + 15x - 4x – 20 3x2 + 11x - 20 The Slip & Slide Method Factoring Trinomials Completing the Square Grouping Method Method of the form ax2 + bx =c Factoring Trinomials where a = 1 andofthe of a, b, c = 1 theGCF form ax2 + bx + c where the GCF of a, b, & c = 1 1) Just like when using the x-box or slip-and-slide methods, find the factors of ac that add up to b. 2) Rewrite the middle term as the sum of the two factors of ac. Group the factors such that each one has a common factor with either a or c. 3) Pull out the common factor(s) of each group. 4) If this method can be used, you will end up with the same thing inside both parentheses. This is one of your factors. 5) The factors that you pulled out of each group form the other factor. 6) Use FOIL to check your factors; make sure that they multiply back to the original quadratic expression. Example: Factor 3x2 + 11x – 20 Step 1 Find factors of the new c that add up to b The numbers that multiply to get -60 and add to get 11 are -4 and 15. On calculator, go to y= and put in -60/x. Hit 2nd Graph and look at the table to find the factors of -60. Find the two that add up to 11. Step 2 Rewrite the middle term as the sum of the Step 3: Pull out the common factor(s) of each two factors of ac. Group the factors such that group. each one has a common factor with either a or c. (3x2 + 15x) – (4x – 20) 3x(x+5) – 4(x+5) Step 4: Pull out the common factor(s) of each group. We put the 15 with the 3 because they share the factor of 3 We put the -4 with the -20 because they share the factor of -4. Step 5: The factors that you pulled out of each group form the other factor. Step 6: Multiply using FOIL to confirm the factors are correct. 3x(x+5) – 4(x+5) 3x(x+5) – 4(x+5) (3x - 4) (x + 5) Since we have the same (x+5) inside both parentheses, this is one of our factors. Since we pulled a 3x out of one group and a -4 out of the other, they form the second factor, 3x-4. 3x2 + 15x - 4x – 20 3x2 + 11x - 20 (3x-4)(x+5) Grouping Method