Introduction to factoring polynomials

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Ex: Factor 2t2 + 5t – 12

1. Is there a GCF?

No

2. Multiply ac 

a =

2

2

and c =

– 12

– 12

2(-12) = - 24

3. Factors of -24 that sum to

+ 5

5

4. Rewrite the middle term

2t2 + 5t – 12 = 2t2 – 3t + 8t – 12

1 – 24  – 23

2 – 12  – 10

3 – 8  – 5

Four-term polynomial . . . Factor by grouping . . .

Close but wrong sign so reverse it

- 3 8  5

Slide 45

2t2 – 3t + 8t – 12

Common factor among the 1st two terms?

 2 t 2 – 3t = t( – )

t

t

t

3

2t

3

Common factor among the 2nd two terms?

4

 8 t – 12 = 4( – )

4

4

3

2t

3

 2t2 – 3t + 8t – 12 = t(2t – 3) + 4(2t – 3)

= (2t – 3)( )

t

+ 4

2

Slide 46

Ex: Factor 9x4 + 18x2 + 8

1. Is there a GCF?

No

2. Multiply ac 

a =

9

9

and c =

8

8

9(8) = 72

3. Factors of 72 that sum to

18

18

4. Rewrite the middle term

9x4 + 18x2 + 8 = 9x4 + 6x2 + 12x2 + 8

1 72  73

3 24  27

6 12  18 

Four-term polynomial . . . Factor by grouping . . .

Bit big  think bigger factors

Slide 47

9x4 + 6x2 + 12x2 + 8

Common factor among the 1st two terms?

 9x4 + 6x2 = 3x2( + )

3x2

3x2

3x2

2

3x2

2

Common factor among the 2nd two terms?

4

 12x2 + 8 = 4( + )

4

4

3

3x2

2

 9x4 + 6x2 + 12x2 + 8 = 3x2(3x2 + 2) + 4(3x2 + 2)

= (3x2 + 2)( + )

3x2

4

3

3x2

Slide 48

Ex: Factor 12x2 – 17xy + 6y2

1. Is there a GCF?

No, but notice two variables

2. Multiply ac 

a =

12x2

12

and c =

6y2

6y2

12x2(6y2) = 72y2

3. Factors of 72x2y2 that sum to

– 17 y

- 17xy

4. Rewrite the middle term

12x2 – 17xy + 6y2 = 12x2 – 8xy – 9xy + 6y2

-1xy -72xy  -73xy

-6xy -12xy  -18xy

-8xy -9xy  -17xy 

Four-term polynomial . . . Factor by grouping . . .

Each factor need a y, both need to be negative

Too big, think bigger factors

Pick one to be the variable

Slide 49

12x2 – 8xy – 9xy + 6y2

Common factor among the 1st two terms?

 12x2 – 8xy = 4x( – )

4x

4x

4x

2y

3x

2y

Common factor among the 2nd two terms?

- 3y

 – 9xy + 6y2 = - 3y( )

-3y

-3y

-2y

3x