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Factoring ax bx c 2 Lesson 2.7 Page 85 More Diamonds #1 30 2 x 11 11 x 15 2 2x2 (2 x 5)( x 3) + 5x + 6x + 15 x (2x + 5) + 3 (2x + 5) (2x + 5)(x + 3) 5 6 • Multiply the coefficient of x2 and the constant. • The product goes on top. • Place the coefficient of x on bottom. • Find the right and left. • Rewrite the trinomial with four terms. • Factor by grouping. Shortcut Way #1 30 2 x 11 11 x 15 2 (2 x 5)( x 3) Step #1: Multiply leading coefficient and constant together and put on top. Step #2: Put coefficient of x on bottom. Step #3: Figure out the left and right numbers to complete the diamond. Step #4: Write the answer from the diamond as if there was a leading coefficient of one. (x+5)(x+6) Step #5: Divide each number in the binomials by the leading coefficient. Reduce the fractions if possible. (x+5/2)(x+6/2) = (x+5/2)(x+3) Step #6: If you are left with a fraction for the number in the binomial move the denominator of the fraction to the coefficient of the variable in the same binomial. (2x+5)(x+3) 5 6 More Diamonds #2 12 (2 x 1)(3x 2) 6 x 77 x 2 2 6x2 + 3x + 4x + 2 3x(2x + 1) + 2 (2x + 1) (2x + 1)(3x + 2) 3 4 • Multiply the coefficient of x2 and the constant. • The product goes on top. • Place the coefficient of x on bottom. • Find the right and left. • Rewrite the trinomial with four terms. • Factor by grouping. Shortcut Way #2 12 6 x 77 x 2 2 (2 x 1)(3x 2) Step #1: Multiply leading coefficient and constant together and put on top. Step #2: Put coefficient of x on bottom. Step #3: Figure out the left and right numbers to complete the diamond. Step #4: Write the answer from the diamond as if there was a leading coefficient of one. (x+3)(x+4) Step #5: Divide each number in the binomials by the leading coefficient. Reduce the fractions if possible. (x+3/6)(x+4/6) = (x+1/2)(x+2/3) Step #6: If you are left with a fraction for the number in the binomial move the denominator of the fraction to the coefficient of the variable in the same binomial. (2x+1)(3x+2) 3 4 More Diamonds #3 -18 2 x -33 x 9 2 (2 x 3)( x 3) 2x2 + 3x - 6x - 9 x (2x + 3) - 3 (2x + 3) (2x + 3)(x - 3) 3 -6 • Multiply the coefficient of x2 and the constant. • The product goes on top. • Place the coefficient of x on bottom. • Find the right and left. • Rewrite the trinomial with four terms. • Factor by grouping. Shortcut Way #3 -18 2 x -33 x 9 2 (2 x 3)( x 3) Step #1: Multiply leading coefficient and constant together and put on top. Step #2: Put coefficient of x on bottom. Step #3: Figure out the left and right numbers to complete the diamond. Step #4: Write the answer from the diamond as if there was a leading coefficient of one. (x+3)(x–6) Step #5: Divide each number in the binomials by the leading coefficient. Reduce the fractions if possible. (x+3/2)(x–6/2) = (x+3/2)(x–3) Step #6: If you are left with a fraction for the number in the binomial move the denominator of the fraction to the coefficient of the variable in the same binomial. (2x+3)(x–3) 3 -6 More Diamonds #4 -12 (2 x 1)(3x 2) 6 x 1 x 2 2 6x2 - 3x + 4x - 2 3x(2x – 1) + 2 (2x – 1) (2x – 1)(3x + 2) -3 4 • Multiply the coefficient of x2 and the constant. • The product goes on top. • Place the coefficient of x on bottom. • Find the right and left. • Rewrite the trinomial with four terms. • Factor by grouping. Shortcut Way #4 -12 6 x 1 x 2 2 (2 x 1)(3x 2) Step #1: Multiply leading coefficient and constant together and put on top. Step #2: Put coefficient of x on bottom. Step #3: Figure out the left and right numbers to complete the diamond. Step #4: Write the answer from the diamond as if there was a leading coefficient of one. (x–3)(x+4) Step #5: Divide each number in the binomials by the leading coefficient. Reduce the fractions if possible. (x–3/6)(x+4/6) = (x–1/2)(x+2/3) Step #6: If you are left with a fraction for the number in the binomial move the denominator of the fraction to the coefficient of the variable in the same binomial. (2x–1)(3x+2) -3 4 #5 A Systematic Approach to Finding the Right & Left • • • • • Multiply the coefficient of x2 and the constant. List all of the factor pairs of the product. Find the pair that add/subtract to yield the middle term. Rewrite the trinomial with four terms. Factor by grouping. 6 x 7 x 20 2 6 x 8 x 15 x 20 2 x( ) 5( ) 2 x(3x 4) 5(3x 4) (3x 4)( 2 x 5) 2 (6)(-20) = -120 1(120) 2(60) 3(40) 4(30) 5(24) 6(20) 8(15) 10(12) If the product is positive, then add the factors. If the product is negative, then subtract the factors. #6 A Systematic Approach to Finding the Right & Left • • • • • Multiply the coefficient of x2 and the constant. List all of the factor pairs of the product. Find the pair that add/subtract to yield the middle term. Rewrite the trinomial with four terms. Factor by grouping. 2 x 13x 20 2 2 x 8 x 5 x 20 2 x( ) 5( ) 2 x( x 4) 5( x 4) ( x 4)( 2 x 5) 2 (2)(20) = 40 1(40) 2(20) 4(10) 5(8) If the product is positive, then add the factors. If the product is negative, then subtract the factors. #7 Try it your favorite way! 2 4 x 4 x 35 Answer: (2x–7)(2x+5) Homework Assignment: Complete # 8 – 14 on notes handout