Introduction to factoring polynomials

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Think of the Distributive Law: a(b+c) = ab + ac  reverse it ab + ac = a(b + c)

Do the terms share a common factor?

What is the GCF(7x2, 3x)?

Recall: GCF(7x2, 3x) = x



7

x

2

+

3

x

=

x

( + )

What’s left?

x

x

Factor out

 7x2 + 3x = x(7x + 3)

Slide 9

Ex: Factor 6a5 – 3a3 – 2a2

Recall: GCF(6a5,3a3,2a2)= a2

6a5 – 3a3 – 2a2 = a2( - - )

a2

a2

a2

3

1

6a3

3a

2

 6a5 – 3a3 – 2a2 = a2(6a3 – 3a – 2)

Slide 10

Your Turn to Try a Few

Slide 11

Ex: Factor x(a + b) – 2(a + b)

Always ask first if there is common factor the terms share . . .

x(a + b) – 2(a + b)

Each term has factor (a + b)

 x(a + b) – 2(a + b)

= (a + b)( – )

(a + b)

(a + b)

x

2

 x(a + b) – 2(a + b) = (a + b)(x – 2)

Slide 12

Ex: Factor a(x – 2) + 2(2 – x)

As with the previous example, is there a common factor among the terms?

Well, kind of . . . x – 2 is close to 2 - x . . . Hum . . .

Recall: (-1)(x – 2) =

- x + 2 = 2 – x

 a(x – 2) + 2(2 – x) =

a(x – 2) + 2((-1)(x – 2))

= a(x – 2) + (– 2)(x – 2)

= a(x – 2) – 2(x – 2)

 a(x – 2) – 2(x – 2) =

(x – 2)( – )

(x – 2)

(x – 2)

a

2

Slide 13

Ex: Factor b(a – 7) – 3(7 – a)

Common factor among the terms?

Well, kind of . . . a – 7 is close to 7 - a

Recall: (-1)(a – 7) =

- a + 7 = 7 – a

 b(a – 7) – 3(7 – a) =

b(a – 7) – 3((-1)(a – 7))

= b(a – 7) + 3(a – 7)

= b(a – 7) +3(a – 7)

 b(a – 7) + 3(a – 7) =

(a – 7)( + )

(a – 7)

(a – 7)

b

3

Slide 14

Your Turn to Try a Few

Slide 15

To factor a polynomial completely, ask

Do the terms have a common factor (GCF)?

Does the polynomial have four terms?

Is the polynomial a special one?

Is the polynomial a difference of squares?

a2 – b2

Is the polynomial a sum/difference of cubes?

a3 + b3 or a3 – b3

Is the trinomial a perfect-square trinomial?

a2 + 2ab + b2 or a2 – 2ab + b2

Is the trinomial a product of two binomials?

Factored completely?

Slide 16

Factor by Grouping