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Diamond Factorization: Simplifying Trinomials with Shortcut Way

Learn to factorize trinomials using the diamond method. Multiply coefficients and constants, work out left and right numbers, and simplify binomials. Follow systematic steps for efficient results.

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Diamond Factorization: Simplifying Trinomials with Shortcut Way

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  1. Lesson 2.7 Page 85

  2. More Diamonds #1 30 • 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. 11 2x2 + 5x + 6x + 15 x (2x + 5) (2x + 5) + 3 (2x + 5) (x + 3) 5 6

  3. Shortcut Way #1 30 11 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

  4. More Diamonds #2 12 • 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. 7 6x2 + 3x + 4x + 2 3x (2x + 1) (2x + 1) + 2 4 (2x + 1) (3x + 2) 3

  5. Shortcut Way #2 12 7 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

  6. More Diamonds #3 -18 • 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. -3 2x2 + 3x - 6x - 9 x (2x + 3) (2x + 3) - 3 -6 3 (2x + 3) (x - 3)

  7. Shortcut Way #3 -18 -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

  8. More Diamonds #4 -12 • 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. 1 6x2 - 3x + 4x - 2 3x (2x – 1) (2x – 1) + 2 (2x – 1) (3x + 2) -3 4

  9. Shortcut Way #4 -12 1 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

  10. 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)(-20) = -120 1(120) 2(60) 3(40) 4(30) 5(24) 6(20) 8(15) 10(12) #5 A Systematic Approach to Finding the Right & Left If the product is positive, then add the factors. If the product is negative, then subtract the factors.

  11. 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)(20) = 40 1(40) 2(20) 4(10) 5(8) #6 A Systematic Approach to Finding the Right & Left If the product is positive, then add the factors. If the product is negative, then subtract the factors.

  12. #7 Try it your favorite way! Answer: (2x–7)(2x+5) Homework Assignment: Complete # 8 – 14 on notes handout

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