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Factoring Polynomials The Diamond Method

Factoring Polynomials The Diamond Method. See if you can discover a pattern in the diamonds below. 10. 6. 4. -8. 5. 2. 2. 3. -1. -4. -2. 4. -5. 7. 5. 2. Fill in the diamonds below. 12. 6. -21. -40. 4. 3. -2. -3. 7. -3. -8. 5. 7. -5. 4. -3.

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Factoring Polynomials The Diamond Method

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  1. Factoring Polynomials The Diamond Method

  2. See if you can discover a pattern in the diamonds below. 10 6 4 -8 5 2 2 3 -1 -4 -2 4 -5 7 5 2

  3. Fill in the diamonds below 12 6 -21 -40 4 3 -2 -3 7 -3 -8 5 7 -5 4 -3

  4. Fill in the diamonds below 24 -12 -20 28 2 12 -1 2 -10 -4 -7 12 14 11 -8 -11

  5. Factoring Polynomials How do you factor a trinomial with leading coefficient of 1? Factor x2 -13x +36

  6. You can use a diamond... Factor x2 -13x + 36 Write the last term here. +36 Now, find factors that will multiply to the top number, and add to the bottom number. -9-4 -13 Write the middle coefficient here The factors are (x – 9)(x – 4)

  7. Factor x2 – 3x – 40 –40 –3 –8 +5 The factors are(x – 8)(x + 5)

  8. Factor x2 + 14x + 24 24 14 12 2 The factors are (x + 12)(x + 2)

  9. Factor x2 + 11x - 12 -12 11 12 -1 The factors are (x + 12)(x - 1)

  10. Factor x2 - 8x - 20 -20 -8 -10 2 The factors are (x - 10)(x + 2)

  11. Factor x2 -11x + 28 28 -11 -4 -7 The factors are (x - 4)(x - 7)

  12. How do you factor a trinomial whose leading coefficient is not 1? Factor 3x2 + 13x + 4 12 1 12 13 (x + 12)(x + 1) = x2 + 13x + 12 What happened?

  13. (x + 12)(x + 1) The diamond method needs help when the leading coefficient is not equal to 1. We must use the fact that the leading coefficientis 3. (x + 12)(x + 1) 3 3

  14. Now, reduce the fractions, if possible. The coefficient of x will be the reduced denominator. 12/3 = 4/11/3 is reduced. (1x + 4)(3x + 1)

  15. Let’s try another. Factor 6x2 + x -15 -90 1 10 -9 (x + 10) (x – 9) , but we must divide. 6 6 Reduce the fractions.

  16. 10/6 = 5/3 -9/6 = -3/2 (3x + 5)(2x - 3) Now we’ll try an extra for experts factoring problem.

  17. Factor 6d2 + 33d – 63 Remember, look for the GCF first... GCF: 3 3(2d2 + 11d – 21) Now, factor the trinomial -42 14 -3 11 Now we have the factors (x+14) (x-3)

  18. Since the leading coefficient of our trinomial is 2, we need to divide by 2. (x + 14) (x - 3) Reduce 2 2 (x +7) (x - 3) = (1x + 7) (2x – 3) 2 1 Our complete factored form is 3(x + 7) (2x – 3)

  19. Additional Practice

  20. Practice 12 • Factor the binomial x + 8x + 12 6 2 • Factor the binomial x - 6x + 5 2 8 • Did you get (x+6)(x+2). 2 5 • Hint a negative sum and positive product means you are multiplying two negative numbers. -5 -1 -6 • Did you get (x-5)(x-1).

  21. Practice 2 • Factor the binomial x + 2x - 8 -8 • Hint a positive sum and negative product means you are multiplying a larger positive number by a smaller negative number. 4 -2 • Factor the binomial x - 3x - 10 2 • Did you get (x+4)(x-2). -10 2 -5 2 • Hint a negative sum and negative product means you are multiplying a larger negative number by a smaller positive number. -3 • Did you get (x-5)(x+2).

  22. 2 • Factor the binomial x - 9 Practice -9 • Hint a zero sum and negative product means you are multiplying a positive number by an equal negative number. 3 -3 0 • Did you get (x+3)(x-3). 2 • This is called the difference of two squares. x is x times x and -9 is 3 times -3. The sum of 3x and -3x equals zero so there is no middle term. 2 • (-x +3)(x+3) would give you -x + 9 which is also the difference of two squares (it could be written as 9- x ). 2

  23. Factoring Completely 3 2 • Factor the binomial 3x + 15x + 18x 2 x + 5x + 6 • You could use a generic rectangle to factor out 3x. 3x • Not so fast…. x2 + 5x + 6 can be further factored using a diamond. 6 3 2 • Your answer should be 3x(x+3)(x+2) 5

  24. Factoring Completely 3 2 • Factor the binomial 4x + 24x + 28x 2 x + 7x + 6 • Did you could use a generic rectangle to factor out 4x. 4x • Now finish factoring 6 6 1 • Your answer should be 4x(x+6)(x+1) 7

  25. Why The Diamond Method Works

  26. Consider F O I L (ax + b)(cx +d) = acx2 + adx + bcx + bd = acx2 + (ad + bc)x + bd acbd ad + bc Now we have the factors (x + ad) (x + bc) bc ad

  27. Since the leading coefficient of our trinomial is ac, we need to divide by ac. (x + bc) (x + ad) Reduce ac ac (x +b) (x - d) = (ax + b) (cx + d) c a

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