See if you can discover a pattern in the diamonds below.

105 2

7

62 3

5

4-1 -4

-5

-8-2 4

2

Fill in the diamonds below

4 3 -2 -3 7 -3 -8 512

7

6

-5

-21

4

-40

-3

Fill in the diamonds below

24

14

-12

11

-20

-8

28

-11122 -1 12 2 -10 -7-4

Factoring Polynomials

How do you factor a trinomial with leading coefficient of 1?

Factor x2 -13x +36

You can use a diamond...

The factors are (x – 9)(x – 4)

Write the middle coefficient here

Write the last term here.

+36

-13

Now, find factors that will multiply to the top number, and add to the bottom number.-9 -4

Factor x2 -13x + 36

Factor x2 – 3x – 40

–40

–3–8 +5

The factors are (x – 8)(x + 5)

Factor x2 + 14x + 24

24

1412 2

The factors are (x + 12)(x + 2)

Factor x2 + 11x - 12

-12

1112 -1

The factors are (x + 12)(x - 1)

Factor x2 - 8x - 20

-20

-8-10 2

The factors are (x - 10)(x + 2)

Factor x2 -11x + 28

28

-11-4 -7

The factors are (x - 4)(x - 7)

How do you factor a trinomial whose leading coefficient is not 1?

Factor 3x2 + 13x + 4

12

1312 1

(x + 12)(x + 1) = x2 + 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

(1x + 4)(3x + 1)

Now, reduce the fractions, if possible. The coefficient of x will be the reduced denominator.

12/3 = 4/1 1/3 is reduced.

Let’s try another.

Factor 6x2 + x -15

-90

110 -9

(x + 10) (x – 9) , but we must divide. 6 6

Reduce the fractions.

(3x + 5)(2x - 3)

10/6 = 5/3 -9/6 = -3/2

Now we’ll try an extra for experts factoring problem.

Factor 6d2 + 33d – 63Remember, look for the GCF first...

GCF: 3 3(2d2 + 11d – 21)

Now, factor the trinomial

-42

1114 -3

Now we have the factors (x+14) (x-3)

(x + 14) (x - 3)

Since the leading coefficient of our trinomial is 2, we need to divide by 2.

2 2

(x +7) (x - 3)1 2

= (1x + 7) (2x – 3)

Our complete factored form is3(x + 7) (2x – 3)

Reduce

Practice

• Factor the binomial x - 6x + 5

2

8

12

6 2

• Did you get (x+6)(x+2).

• Factor the binomial x + 8x + 12

2

-6

5

-5 -1• Hint a negative sum and positive product means you are multiplying two negative numbers.

• Did you get (x-5)(x-1).

Practice

• Factor the binomial x - 3x - 10

2

2

-8

4 -2

• Did you get (x+4)(x-2).

• Factor the binomial x + 2x - 8

2

-3

-10

-5 2

• Hint a positive sum and negative product means you are multiplying a larger positive number by a smaller negative number.

• Did you get (x-5)(x+2).

• Hint a negative sum and negative product means you are multiplying a larger negative number by a smaller positive number.

Practice2

-9

0

3 -3

• Did you get (x+3)(x-3).

• Factor the binomial x - 9

• Hint a zero sum and negative product means you are multiplying a positive number by an equal negative number.

• 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

2

Factoring Completely2

5

6

3 2

• Factor the binomial 3x + 15x + 18x

• Not so fast…. x2 + 5x + 6 can be further factored using a diamond.

3

3x

x + 5x + 6 2

• You could use a generic rectangle to factor out 3x.

• Your answer should be 3x(x+3)(x+2)

Factoring Completely2

7

6

6 1

• Factor the binomial 4x + 24x + 28x

• Now finish factoring

3

4x

x + 7x + 62

• Did you could use a generic rectangle to factor out 4x.

• Your answer should be 4x(x+6)(x+1)

Consider

(ax + b)(cx +d) = acx2 + adx + bcx + bdF O I L

= acx2 + (ad + bc)x + bd

acbd

ad + bc

bc adNow we have the factors (x + ad) (x + bc)

(x + bc) (x + ad)

Since the leading coefficient of our trinomial is ac, we need to divide by ac.

ac ac

(x +b) (x - d)a c

= (ax + b) (cx + d)

Reduce