7/23/2019 Special Products and Factoring Strategies

http://slidepdf.com/reader/full/special-products-and-factoring-strategies 1/10

Special Products and Factoring Strategies

Review of Three Special Products

Recall the three special products:

1. Difference of Squares

  x2 - y2  = (x - y) (x + y)

2. Square of Su

  x2 + 2xy + y2  = (x + y)2 

!. Square of Difference

  x2 - 2xy + y2  = (x - y)2

Special Products Involving Cubes

"ust as there is a difference of squares forula# there is also a difference of cu$esforula.

%. x! - y! = (x - y) (x2 + xy + y2)

Proof:

&e use the distri$uti'e la on the riht hand side

  x (x2 + xy + y2) - y (x2 + xy + y2)

= x! + x2y + xy2 - x2y - xy2 - y!

 

7/23/2019 Special Products and Factoring Strategies

http://slidepdf.com/reader/full/special-products-and-factoring-strategies 2/10

*. o co$ine li,e ters to et

x! - y!

 ext# e state the su of cu$es forula. 

%   x! + y!  = (x + y)(x2 - xy + y2)

Exercise

ro'e the su of cu$es equation (quation *)

Using the Special Product Forulas for Factoring

 

Exaples:

/actor the folloin

1. !0x2 - %y2  = (0x - 2y) (0x + 2y)  otice that there only to ters.

2. !x! - 12x2 + 12x = !x (x2 - %x + %) Ree$er to pull the / out first.

= !x(x -2)2 

!. x0 - 0% = (x! - 3) (x! + 3)

= (x - 2) (x2 + 2x + %) (x + 2) (x2 - 2x + %)

 

Exercises:

/actor the folloin

7/23/2019 Special Products and Factoring Strategies

http://slidepdf.com/reader/full/special-products-and-factoring-strategies 3/10

4. %*a! $ - 25a$! 

6. 0%x0 - 10x! + 1

. x2 + 2xy + y2 - 31

D. x12 - y12  (hallene ro$le)

Factoring Strategies

• 4lays pull out the / first

• 7oo, for special products. 8f there are only to ters then loo, for su of

cu$es or difference of squares or cu$es. 8f there are three ters# loo, for

squares of a difference or a su.

• 8f there are three ters and the first coefficient is 1 then use siple trinoial

factorin.

• 8f there are three ters and the first coefficient is not 1 then use the 4 ethod.

• 8f there are four ters then try factorin $y roupin.

 

7/23/2019 Special Products and Factoring Strategies

http://slidepdf.com/reader/full/special-products-and-factoring-strategies 4/10

Exercises

4. 45a3b - 20ab3 

6. 0%x0 - 10x! + 1

. x2 + 2xy + y2 - 31

D. x12 - y12  (hallene ro$le)

Factoring Strategies

• 4lays pull out the / first

• 7oo, for special products. 8f there are only to ters then loo, for su of

cu$es or difference of squares or cu$es. 8f there are three ters# loo, for

squares of a difference or a su.

• 8f there are three ters and the first coefficient is 1 then use siple trinoial

factorin.

•8f there are three ters and the first coefficient is not 1 then use the 4 ethod.

• 8f there are four ters then try factorin $y roupin.

 

Exercises

7/23/2019 Special Products and Factoring Strategies

http://slidepdf.com/reader/full/special-products-and-factoring-strategies 5/10

4. x! - x

6. x2

- 9x - !5

. 0a2 $ - %3a$ - 92a + !0

D. %x2 - !0xy + 31y2

. *a% $! +

1535a

/. 2x2 + *x - 12

. *x

!

+ %5

;. x! + !x2 - %x - 12

Factoring in Algebra

Factors

 u$ers ha'e factors:

7/23/2019 Special Products and Factoring Strategies

http://slidepdf.com/reader/full/special-products-and-factoring-strategies 6/10

4nd expressions (li,e x!"#x"$) also ha'e factors:

Factoring

/actorin (called <Factorising< in the >) is the process of finding the factors:

/actorin: /indin hat to ultiply toether to et an expression.

8t is li,e <splittin< an expression into a ultiplication of sipler expressions.

xaple: factor 2y+66oth 2y and 0 ha'e a coon factor of 2:

•

2y is 2 × y• 6 is 2 × 3

 

So you can factor the hole expression into:

2y+6 = 2(y+3)So# 2y+0 has $een <factored into< 2 and y+!

/actorin is also the opposite of xpandin:

7/23/2019 Special Products and Factoring Strategies

http://slidepdf.com/reader/full/special-products-and-factoring-strategies 7/10

Common Factor

8n the pre'ious exaple e sa that 2y and 0 had a coon factor of !

6ut to do the ?o$ properly a,e sure you ha'e the highest coon factor# includin any 'aria$les

xaple: factor 3y2+12y/irstly# ! and 12 ha'e a coon factor of 3.

So you could ha'e:

3y2+12y = 3(y2+4y)6ut e can do $etter@

3y2 and 12y also share the 'aria$le y.

Aoether that a,es 3y:

• 3y2 is 3y × y

• 12y is 3y × 4

 

So you can factor the hole expression into:

3y2+12y = 3y(y+4) 

hec,: $%&%"#' ( $% ) % " $% ) # ( $%!"*!%

 

More Complicated Factoring

 

Factoring Can e !ard "

Ahe exaples ha'e $een siple so far# $ut factorin can $e 'ery tric,y.

6ecause you ha'e to fiure what got ultiplied to produce the expression you are i'en@

8t can $e li,e tryin to find out hat inredients ent

into a ca,e to a,e it so delicious. 8t is soeties not

o$'ious at all@

7/23/2019 Special Products and Factoring Strategies

http://slidepdf.com/reader/full/special-products-and-factoring-strategies 8/10

 

#$perience !elps

6ut the ore experience you et# the easier it $ecoes.

Example: Factor 4x2 - 9;... 8 canBt see any coon factors.

6ut if you ,no your Special 6inoial roducts you iht see it as the +difference of s,uares+:

6ecause #x! is &!x'!# and - is &$'!#

so e ha'e:

4$2 % & = (2$)2 % (3)2

4nd that can $e produced $y the difference of squares forula:

(a+b)(a%b) = a2 % b2

&here a is 2x# and b is !.

So let us try doin that:

(2$+3)(2$%3) = (2$)2 % (3)2 = 4$2 % &Ces@

 

So the factors of #x! . - are &!x"$' and &!x.$':

Ans'er 4$2 % & = (2$+3)(2$%3)

;o can you learn to do that 6y ettin lots of practice# and ,noin <8dentities<@

 

emember t*ese dentities

;ere is a list of coon <8dentities< (includin the +difference of s,uares+ used a$o'e).

8t is orth ree$erin these# as they can a,e factorin easier.

7/23/2019 Special Products and Factoring Strategies

http://slidepdf.com/reader/full/special-products-and-factoring-strategies 9/10

a2 % b2  = (a+b)(a%b)

a2 + 2ab + b2  = (a+b)(a+b)

a2 % 2ab + b2  = (a%b)(a%b)

a3 + b3  = (a+b)(a2%ab+b2)

a3 % b3  = (a%b)(a2+ab+b2)

a3

+3a2

b+3ab2

+b3

 = (a+b)3

a3%3a2b+3ab2%b3  = (a%b)3

Ahere are any ore li,e those# $ut those are the siplest ones.

Ad,ice

Ahe factored for is usually $est.

&hen tryin to factor# follo these steps:

• </actor out< any coon ters

• See if it fits any of the identities# plus any ore you ay ,no

• >eep oin till you canBt factor any ore

Cou can also use coputers@ Ahere are oputer 4le$ra Systes (called <4S<) such as  Axiom,

 Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce and any ore that are ood at factorin.

More #$amples

8 said that experience helps# so here are ore exaples to help you on the ay:

7/23/2019 Special Products and Factoring Strategies

http://slidepdf.com/reader/full/special-products-and-factoring-strategies 10/10

Example: w4 - 164n exponent of % Eay$e e could try an exponent of 2:

'4 % 16 = ('2)2 % 42

Ces# it is the difference of squares

'4

 % 16 = ('2

+ 4)('2

% 4)4nd <(2 - %)< is another difference of squares

'4 % 16 = ('2 + 4)(' + 2)(' % 2)Ahat is as far as 8 can o (unless 8 use iainary nu$ers)

Example: 3u4 - 24uv3

Reo'e coon factor <!u<:

3-4 % 24-,3 = 3-(-3 % .,3)Ahen a difference of cu$es:

3-4 % 24-,3 = 3-(-3 % (2,)3)

= 3-(-%2,)(-2+2-,+4,2)Ahat is as far as 8 can o.

Example: z3 - z2 - 9z + 9

/ry 0actoring t*e 0irst t'o and second t'o separately

2(%1) % &(%1)o' (%1) is on bot* so let -s -se t*at

(2%&)(%1)And 2%& is a di00erence o0 s-ares

(%3)(+3)(%1)