Introduction to factoring polynomials

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If the polynomial has four terms, consider factor by grouping

Factor out the GCF from the first two terms

Factor out the GCF from the second two terms (take the negative sign if minus separates the first and second groups)

If factor by grouping is the correct approach, there should be a common factor among the groups

Factor out that GCF

Check by multiplying using FOIL

Slide 17

Ex: Factor 6a3 + 3a2 +4a + 2

Notice 4 terms . . . think two groups: 1st two and 2nd two

Common factor among the 1st two terms?

 6a3 + 3a2 = 3a2( + )

GCF(6a3, 3a2)

= 3a2

3a2

3a2

2a

1

2a

1

Common factor among the 2nd two terms?

GCF(4a, 2)

= 2

 4a + 2 = 2( + )

2

2

2

1

2a

1

Now put it all together . . .

Slide 18

6a3 + 3a2 +4a + 2 =

3a2(2a + 1) + 2(2a + 1)

Four terms  two terms. Is there a common factor?

Each term has factor (2a + 1)

3a2(2a + 1) + 2(2a + 1)

= (2a + 1)( + )

(2a + 1)

(2a + 1)

3a2

2

6a3 + 3a2 +4a + 2 =

(2a + 1)(3a2 + 2)

Slide 19

Ex: Factor 4x2 + 3xy – 12y – 16x

Notice 4 terms . . . think two groups: 1st two and 2nd two

Common factor among the 1st two terms?

 4x2 + 3xy = x( + )

GCF(4x2, 3xy)

= x

x

x

4x

3y

4x

3y

Common factor among the 2nd two terms?

GCF(-12y, - 16x)

= -4

 -12y – 16x = - 4( )

-4

-4

3y

4x

3y

+ 4x

Now put it all together . . .

Slide 20

4x2 + 3xy – 12y – 16x =

x(4x + 3y) – 4(4x + 3y)

Four terms  two terms. Is there a common factor?

Each term has factor (4x + 3y)

x(4x + 3y) – 4(4x + 3y)

= (4x + 3y)( )

(4x + 3y)

(4x + 3y)

x

– 4

4x2 + 3xy – 12y – 16x =

(4x + 3y)(x – 4)

Slide 21

Ex: Factor 2ra + a2 – 2r – a

Notice 4 terms . . . think two groups: 1st two and 2nd two

Common factor among the 1st two terms?

 2ra + a2 = a( + )

GCF(2ra, a2)

= a

a

a

2r

a

Common factor among the 2nd two terms?

GCF(-2r, - a)

= -1

 -2r – a = - 1( )

-1

-1

2r

+ a

Now put it all together . . .

Slide 22

2ra + a2 –2r – a =

a(2r + a) – 1(2r + a)

Four terms  two terms. Is there a common factor?

Each term has factor (2r + a)

a(2r + a) – 1(2r + a)

= (2r + a)( )

(2r + a)