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 below12 6
-21
-40 4 3 -2 -3 7 -3 -8 5 7 -5 4 -3

4. Fill in the diamonds below24
-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. Factor x2 -13x + 36 You can use a diamond... +36 -9 -4 -13
Write the last term here.
+36
Now, find factors that will multiply to the top number, and add to the bottom number.
-13
Write the middle coefficient here
The factors are (x – 9)(x – 4)

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

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

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

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

11. The factors are (x - 4)(x - 7)
Factor x2 -11x + 28 28
-11
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 12 1 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 coefficient is 3.
(x + 12)(x + 1)

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

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

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

17. Now, factor the trinomial
Factor 6d2 + 33d – 63
Remember, look for the GCF first...
GCF: (2d2 + 11d – 21)
Now, factor the trinomial
-42 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) 1 2
Our complete factored form is
3(x + 7) (2x – 3)

19. Additional Practice

20. Practice 12 Factor the binomial x + 8x + 12 6 2 8
Did you get (x+6)(x+2). 2
Factor the binomial x x + 5 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 Factor the binomial x + 2x - 8 -8 4 -2 2
Hint a positive sum and negative product means you are multiplying a larger positive number by a smaller negative number. 4 -2 2
Did you get (x+4)(x-2).
-10 2
Factor the binomial x - 3x - 10 -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. Practice Factor the binomial x - 9 -9 3 -3 Did you get (x+3)(x-3).2
Factor the binomial x -9
Hint a zero sum and negative product means you are multiplying a positive number by an equal negative number. 3 -3
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 x
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 Now finish factoring 6 6 13 2
Factor the binomial 4x + 24x + 28x 2
x x
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 = acx2 + adx + bcx + bd = acx2 + (ad + bc)x + bd 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) ac
Reduce ac
(x +b) (x - d)
= (ax + b) (cx + d) a c