Special Factoring Formulas

By Mr. Richard Gill

and Dr. Julia Arnold

In this presentation we will be studying special factoring formulas for:A) Perfect Square Trinomials B) The difference of two squaresC) The sum or difference of two cubes.

A) Perfect Square Trinomials

Perhaps you remember the formula for squaring a binomial:

The right side of each of the above equations is called a “perfect square trinomial.” You can factor a perfect square trinomial by trial and error but, if you recognize the form, you can do the job easier and faster by using the factoring formula.

222

222

bab2aba

bab2aba

)(

)(

To factor a perfect square trinomial you look for certain clues:1. Is the first term a perfect square of the forma2 for some expression a?

2. Is the last term a perfect square of the formb2 for some expression b?

3. Is the middle term plus or minus 2 times the square root of a2 times the square root of b2?

1x2x2

For example: Is the following a perfect square trinomial?

1x2x2

Is this term a perfect square? Yes.

Is this term a perfect square? Yes.

Is the middle term plus or minus 2 times the square root of x2 times the square root of 12?

x times 1 times 2 = 2x yes.

Then this is a perfect square and factors into(x + 1)2

See if you can recognize perfect square trinomials.

1. 25x10x2 Yes, this is (x + 5)2

2. 4x4x2 Yes, this is (x + 2)2

Yes, this is (3y - 2)23. 4y12y9 2

y122y32

24y3y9 2

...

36y12y2 4.

No, -36 is not a perfect square

T h i s p o i n t c o n c e r n i n g t h e m i d d l e t e r m i s i m p o r t a n t . C o n s i d e r t h e f o l l o w i n g e x a m p l e :

F a c t o r .91 44 2 xx T h e fi r s t t e r m a n d t h e l a s t t e r m a r e b o t h p e r f e c t s q u a r e s b u t t h e m i d d l e t e r m i s n o t t r u e t o f o r m . 2 ( 2 ) ( 3 ) = 1 2 o r – 1 2 s h o u l d b e t h e m i d d l e t e r m .

B y t r i a l a n d e r r o r .9149144 2 xxxx W e c o u l d h a v e a p e r f e c t s q u a r e t r i n o m i a l b y c h a n g i n g t h e m i d d l e t e r m :

22 3232329124 xxxxx .

Is the middle term 2 times the square root of a2 times the square root of b2

Try the following four problems. Watch for thepattern. Do all four before you check theanswers on the next slide.

49284

64489

3613

6416

2

2

2

2

xx

xx

xx

xx

T h e s o l u t i o n s a r e :

22

22

2

22

72727249284

83838364489

493613

8886416

xxxxx

xxxxx

xxxx

xxxxx

T h e s e c o n d p r o b l e m w a s n o t a p e r f e c t s q u a r eb u t w a s f a c t o r e d a s a s i m p l e t r i n o m i a l . T h eo t h e r s w e r e p e r f e c t s q u a r e t r i n o m i a l s .

Try four more. Watch for the pattern for perfect square trinomials: the first term and the last term are perfect squares and the middle term is plus or minus 2ab. If your trinomial is not a perfect square, proceed by trial and error.

1610

169

254016

10020

2

2

2

2

xx

xx

xx

xx

Do all four before you go to the next slide.

H e r e a r e t h e s o l u t i o n s .

)10(21610

131313169

545454254016

10101010020

2

22

22

22

xxxx

xxxxx

xxxxx

xxxxx

T h e f i r s t t h r e e t r i n o m i a l s w e r e p e r f e c t s q u a r e s . N o t e t h a t i n e a c h c a s e , t w i c e t h e p r o d u c t o f t h e t e r m s i n t h e a n s w e r i s e x a c t l y t h e m i d d l e t e r m o f t h e t r i n o m i a l . T h e l a s t t r i n o m i a l i s n o t a p e r f e c t s q u a r e t r i n o m i a l .

B) The Difference of Two Squares

In the difference of two squares

This side is expanded

yxyxyx 22 This side is factored.

x (in the factored side) is the square root of x2 and y (in the factored side) is the square root of y2

Factoring the difference of two squares: some examples.

Note: The sum of two squares is not factorable.

36x2 Why does this binomial not factor?

Suppose it did. What would be the possible factors?

36x12x6x6x 2

3666

3612662

2

xxx

xxxx

No

No

There are no other possibilities, thus it doesn’t factor. In general, the sum of two squares does not factor.

No

Factoring the difference of two squares: Write out your answers before you go to the next page.

The solutions are:

1717149

98988164

5353259

1010100

2

2

2

2

xxx

xxx

yyy

xxx

Factoring the difference of two squares: Write down your solutions before going to the next page.

Factoring the difference of two squares: Here are the solutions.

C) The sum or difference of two cubes

F i n a l l y , w e l o o k a t t w o s p e c i a l p r o d u c t s i n v o l v i n g c u b e s . T h e s u m o f t w o c u b e s : 2233 babababa T h e d i f f e r e n c e o f t w o c u b e s : 2233 babababa

1000 is a perfect cube since 1000 = 103

125 is a perfect cube since 125 = 53

64 is a perfect cube since 64 = 43

8 is a perfect cube since 8 = 23

1 is a perfect cube since 1 = 13

In order to use these two formulas, you must be able to recognize numbers that are perfect cubes.

The following can be factored as the difference of two cubes: letting a = 2x and b = 3

278 3 x

2233 babababa

9643232 233 xxxx

Let’s check with multiplication to see if the factors are correct:

278

27181218128964323

2232

x

xxxxxxxx

F o r a n o t h e r e x a m p l e , t h e e x p r e s s i o n 643 x c a n b ef a c t o r e d a s t h e s u m o f t w o c u b e s u s i n g

))(( 2233 babababa w i t h xa a n d4b .

164464 23 xxxx

T o c h e c k t h i s a n s w e r w i t h m u l t i p l i c a t i o n ,

.64

6416416416443

2232

x

xxxxxxxx

A way to remember the formula

a3 + b3 = ( )( )a + bCube root of a and b with same sign

Clues to remember the formula:1. Cubes always factor into a binomial times a trinomial.

2. The binomial in the factored version always contains the cube roots of the original expression with the same sign that was used in the original expression.

a3 - b3 = ( )( )a - bCube root of a and b with same sign

A way to remember the formula

a3 + b3 = ( )( )a + b a2

1)Square first term

-ab

2)Find the product of both terms and change the sign i.e. a(b) =ab change the sign = -ab

+b2

3)Square last term i.e. last term is b2

Next you use the binomial to build your trinomial:

a3 - b3 = ( )( )A binomial times a trinomial

a - bCube root of a and b with same sign

a2

1)Square first term

+ab

2)Find the product of both terms and change the sign

+b2

3)Square last term

The difference of two cubes:

Cube Numbers you will findin problems.

13=123=8

33=27

43=64

53=125 etc3=etc

Difference of two cubes

8x3 - 125= ( )( )A binomial times a trinomial

2x - 5Cube root of 1st and 2nd term with same sign

4x2

1)Square first term

+10x

2)Find the product of both terms and change the sign

+25

3)Square last term

Build trinomialwith binomial.

Sum or Differenceof two cubes

64x3 + 1= ( )( )A binomial times a trinomial

4x + 1 16x2 -4x + 1

Build trinomial with binomial.

Now try the following four problems on yourown. The answers are on the next page.Factor each of the following as the sum oftwo cubes or as the difference of two cubes.

1000

64

1258

27

3

3

3

3

x

y

x

x

Here are the solutions to the four factoring problems with the formula substitutions for a and b at the end.

10,

10010101000

,4

416464

5,2

25104521258

3,

93327

23

23

23

23

bxa

xxxx

yba

yyyy

bxa

xxxx

bxa

xxxx

Now try four more problems on your own.The answers are on the next page. Factoreach of the following as the sum of two cubesor as the difference of two cubes.

3

3

3

3

12527

164

125

2764

x

x

x

x

H e r e a r e t h e s o l u t i o n s t o t h e f o u r f a c t o r i n gp r o b l e m s o n t h e p r e v i o u s p a g e .

23

23

23

23

251595312527

141614164

2555125

91216342764

xxxx

xxxx

xxxx

xxxx

T his is t he end of t he unit on special f act or ing. A t t his point you should move on t o t he pract ice t est f or special f act or ing. For t he exam, you will need t o memor ize the f ollowing f ormulas:

2233

2233

22

222

222

2

2

babababa

babababa

bababa

bababa

bababa