See if you can discover a pattern in
the diamonds below.
10
5
6
2
7
2
4
3
5
-1
-5
-4
-2
-8
2
4
Fill in the diamonds below
12
4
7
6
3
-2
-5
-3
-21
7
-3
4
-40
-8
-3
5
Fill in the diamonds below
2
24
14
12
-12
-1 12
11
-20
2 -10
-8
-4
28
-11
-7
Factoring Polynomials
How do you factor a trinomial
with leading coefficient of 1?
Factor x2 -13x +36
2
x
Factor -13x + 36
You can use a diamond...
Write the last term here.
+36
-9 -4
-13
Now, find factors that will
multiply to the top number, and
add to the bottom number.
Write the middle coefficient here
The factors are (x – 9)(x – 4)
Factor
2
x
– 3x – 40
–40
–8 +5
–3
The factors are (x – 8)(x + 5)
Factor
2
x
+ 14x + 24
24
12
14
2
The factors are (x + 12)(x + 2)
Factor
2
x
+ 11x - 12
-12
12
11
-1
The factors are (x + 12)(x - 1)
Factor
2
x
- 8x - 20
-20
-10
-8
2
The factors are (x - 10)(x + 2)
Factor
2
x
-11x + 28
28
-4
-7
-11
The factors are (x - 4)(x - 7)
How do you factor a trinomial whose
leading coefficient is not 1?
Factor
2
3x
12
+ 13x + 4
12
1
13
2
x
(x + 12)(x + 1) = + 13x + 12
What happened?
(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)
3
3
Now, reduce the fractions, if
possible. The coefficient of x will
be the reduced denominator.
12/3 = 4/1
1/3 is reduced.
(1x + 4)(3x + 1)
Let’s try another.
Factor 6x2 + x -15
10
-90
1
-9
(x + 10) (x – 9) , but we must divide.
6
6
Reduce the fractions.
10/6 = 5/3
-9/6 = -3/2
(3x + 5)(2x - 3)
Now we’ll try an extra for experts
factoring problem.
2
Factor 6d + 33d – 63
Remember, look for the GCF first...
2
GCF: 3
3(2d + 11d – 21)
Now, factor the trinomial
-42
14 -3
11
Now we have the factors (x+14) (x-3)
Since the leading coefficient of our
trinomial is 2, we need to divide by 2.
(x + 14) (x - 3)
2
2
(x +7) (x - 3)
2
1
Reduce
= (1x + 7) (2x – 3)
Our complete factored form is
3(x + 7) (2x – 3)
Practice
12
• Factor the binomial
2
x + 8x + 12
6
2
8
• Did you get (x+6)(x+2).
2
• Factor the binomial x - 6x + 5
• Hint a negative sum and positive
product means you are multiplying
two negative numbers.
• Did you get (x-5)(x-1).
5
-5
-1
-6
Practice
2
• Factor the binomial x + 2x - 8
• Hint a positive sum and negative product
means you are multiplying a larger
positive number by a smaller negative
number.
-8
4
-2
2
• Did you get (x+4)(x-2).
-10
2
• Factor the binomial x - 3x - 10
• Hint a negative sum and negative
product means you are multiplying
a larger negative number by a
smaller positive number.
-5
2
-3
• Did you get
(x-5)(x+2).
2
Practice
• Factor the binomial x - 9
-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
2
written as 9- x ).
Factoring Completely
3
2
• Factor the binomial 3x + 15x + 18x
• You could use a
generic rectangle to
factor out 3x.
x
2
+ 5x
+6
3x
• Not so fast…. x2 + 5x + 6 can be
further factored using a diamond.
• Your answer should be
3x(x+3)(x+2)
6
3
2
5
Factoring Completely
3
2
• Factor the binomial 4x + 24x + 28x
• Did you could use a
generic rectangle to
factor out 4x.
x
2
+ 7x
+6
4x
• Now finish factoring
6
6
• Your answer should be
4x(x+6)(x+1)
1
7
Consider
F
O
I
L
(ax + b)(cx +d) = acx2 + adx + bcx + bd
= acx2 + (ad + bc)x + bd
acbd
bc
ad
ad + bc
Now we have the factors
(x + ad) (x + bc)
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)
c
a
= (ax + b) (cx + d)