Factoring the sum and difference of two cubes

ByDr. J. Arnold

Look at two special products involving cubes. The sum of two cubes: 2233 babababa The difference of two cubes: 3 3 2 2a b a b a ab b

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

This is the end of the unit on f actoring the sum and diff erence of cubes. The f ollowing is a list of f ormulas f or perfect squares, diff erence of two squares, and sum or diff erence of cubes.

.

2233

2233

22

222

222

2

2

babababa

babababa

bababa

bababa

bababa