factoring

Copyright © 2011 Pearson Education, Inc. Slide -1

Factoring is the process of finding polynomials, called factors, whose product equals a given polynomial.

A polynomial that cannot be written as a product of polynomials with integer coefficients is a prime or irreducible polynomial.

A polynomial is factored completely when it is written as a product of prime polynomials with integer coefficients.


Example Factor out the greatest common factor.

(a) (b)

Solution

(a)

(b)

5 29y y 26 8 12x t xt t

5 2 2 3 2 2 39 (9 ) (1) (9 1)y y y y y y y

2 26 8 12 2 (3 4 6)x t xt t t x x


Example Factor each polynomial by grouping.

(a) (b)

Solution

(a)

2 27 3 21mp m p 3 24 2 2 1x x x

2 2 2 2

2 2

2

7 3 21 ( 7 ) (3 21)

( 7) 3( 7)

( 7)( 3)

mp m p mp m p

m p p

p m


Solution

(b) 3 2 2

2

4 2 2 1 2 (2 1) 1(2 1)

(2 1)(2 1)

x x x x x x

x x


Example Factor each trinomial.

(a)

(b)

24 11 6y y

26 7 5p p


Solution (a) We must find integers a, b, c, d such that

By FOIL, we see that ac = 4 and bd = 6. Thus a and c

are 1 and 4 or 2 and 2.

Since the middle term is negative, consider only negative values for b and d. The possibilities are

–1 and –6 or –2 and –3.

24 11 6 ( )( )y y ay b cy d


Solution (a) Try these combinations of factors

The last trial gives the correct factorization.

2

2

2

(2 1)(2 6) 4 14 6

(2 2)(2 3) 4 10 6

( 2)(4 3) 4 11 6

y y y y

y y y y

y y y y


Solution (b) Try the various combinations of factors

The last trial gives the correct factorization.

2

2

2

(2 5)(3 1) 6 13 5

(2 5)(3 1) 6 13 5

(3 5)(2 1) 6 7 5

p p p p

p p p p

p p p p


Perfect Square Trinomials

2 2 2

2 2 2

2 ( )

2 ( )

x xy y x y

x xy y x y


Example Factor each polynomial.

(a) (b)

Solution

(a)

(b)

2 216 40 25p pq q

2 2 4 2 2 2 2

2 2

169 104 16 (13 ) 2(13 )(4 ) (4 )

(13 4 )

x xy y x x y y

x y

2 2 2 2

2

16 40 25 (4 ) 2(4 )(5 ) (5 )

(4 5 )

p pq q p p q q

p q

2 2 4169 104 16x xy y


Solution

210(2 1) 19(2 1) 15

[5(2 1) 3][2(2 1) 5]

(10 5 3)(4 2 5)

(10 2)(4 7)

2(5 1)(4 7) .

a a

a a

a a

a a

a a


Difference of Squares

2 2 ( )( )x y x y x y



(a) (b)

Solution

(a)

(b)

24 9m 2 46 9x x y

2 2 24 9 (2 ) 3 (2 3)(2 3)m m m m

2 4 2 2 2

2 2

2 2

6 9 ( 3) ( )

[( 3) ][( 3) ]

( 3 )( 3 )

x x y x y

x y x y

x y x y


Difference and Sum of Cubes

Difference of cubes

Sum of cubes

3 3 2 2( )( )x y x y x xy y

3 3 2 2( )( )x y x y x xy y



(a) (b)

Solution

(a)

3 27x 3 364m n

3 3 3

2 2

2

27 3

( 3)( 3 3 )

( 3)( 3 9)

x x

x x x

x x x


Solution

(b) 3 3 3 3

2 2

2 2

64 (4 )

( 4 )[ (4 ) (4 ) ]

( 4 )( 4 16 )

m n m n

m n m m n n

m n m mn n


Example Factor the polynomial

Solution Replacing 2a – 1 with m and

factoring gives

Now, replace m with 2a – 1 in the factored form and

simplify.

210(2 1) 19(2 1) 15 .a a

210 19 15 (5 3)(2 5)m m m m