Introduction to factoring polynomials

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– 2y

 12x2 – 8xy – 9xy + 6y2 = 4x(3x – 2y) – 3y(3x – 2y)

= (3x – 2)( )

4x

– 3y

3

3x

Slide 50

Your Turn to Try a Few

Slide 51

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 52

Ex: Factor x3 + 3x2 – 4x – 12

1. Is there a GCF?

No

2. Notice four terms

 grouping

Common factor among the 1st two terms?

x2

 x3 + 3x2 = x2( + )

x2

x2

x

x

3

Common factor among the 2nd two terms?

- 4

 – 4x – 12 = – 4( )

- 4

- 4

3

x

+ 3

 x3 + 3x2 - 4x – 12 = x2(x + 3) – 4(x + 3)

= (x + 3)( )

x2

– 4

Slide 53

Cont: we have (x + 3)(x2 – 4)

But are we done?

No. We have to make sure we factor completely.

Is (x + 3) prime?  can x + 3 be factored further?

No . . . It is prime

What about (x2 – 4)?

Recognize it?

Difference of Squares

x2 = (x)2

4 = (2)2

 x2 – 4 = (x)2 – (2)2

= (x – 2)(x + 2)

Therefore x3 + 3x2 – 4x – 12 = (x + 3)(x2 – 4)

= (x + 3)(x – 2)(x + 2)

Slide 54

Your Turn to Try a Few

Slide 55

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 56

Special Polynomials

Is the polynomial a sum/difference of cubes?

a3 + b3 = (a + b)(a2 - ab + b2)

a3 – b3 = (a - b)(a2 + ab + b2)

Slide 57

Ex: Factor 8p3 – q3

Notice the terms are both perfect cubes

8p3 = (2p)3

q3 = (q)3