# Lesson Factoring the Sum and Difference of Cubes

Algebra ->  Algebra  -> Polynomials-and-rational-expressions -> Lesson Factoring the Sum and Difference of Cubes      Log On

 Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help! Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations!

 Algebra: Polynomials, rational expressions and equations Solvers Lessons Answers archive Quiz In Depth

This Lesson (Factoring the Sum and Difference of Cubes) was created by by rapaljer(4667)  : View Source, Show
About rapaljer: Retired Professor of Mathematics from Seminole State College of Florida after 36 years.

This text was imported from http://www2.seminolestate.edu/rrapalje/BasicAlgebra/Basic%20One%20Step%20Ch%202/2.07%20Factoring%20by%20Grouping.htm by its author.

2.07  Factoring by Grouping

Difference and Sum of Cubes

from Basic Algebra: One Step at a Time © 2002-2011

P. 173-180

Dr. Robert J. Rapalje

Seminole State College of Florida

 To see selected solutions in Living C O L O R  click here!

This lesson introduces the technique of factoring by grouping, as well as factoring the sum and difference of cubes.  Factoring by grouping builds on the ideas that were presented in the section on factoring the common factor.  While there are many different types of grouping, as you will learn in higher algebra courses, all of the grouping problems in this book involve four terms, and they work by grouping the first two terms and the second two terms together.   If you do it right, a common factor will always emerge!  Remember, the method of grouping is one of trial and error.  As always, there is no substitute for practice and experience.  The second half of this lesson is the sum and difference of cubes, which may be optional to your curriculum this first level of algebra.  Remember, if you can learn this topic now, it will help you later.

FACTORING BY GROUPING

EXAMPLE 1.             Factor  x3 + 2x2 + 8x + 16

Solution:         There are no common factors to all four terms.  It is not a trinomial, and nothing discussed so far works to factor this.  So, try grouping the first two terms together, and the last two terms together, and factor out the common factor within each grouping as follows:

(x3 + 2x2) + (8x + 16) =  x2 (x + 2)+ 8(x + 2)

Notice that there is a common factor of (x+2) that can be factored out:

=  (x + 2)( x2 + 8)

EXAMPLE 2.            Factor  xy 4y + 3x 12

Solution:         Again, there are no common factors, and this is not a trinomial.  Group the first two and the last two terms together, and factor out the common factor from each grouping:

(xy 4y) + (3x 12) =  y(x 4) + 3(x 4)

Now, take out the common factor, which is (x 4):

=  (x 4)(y + 3)

EXAMPLE 3.             Factor  xy 4y 3x + 12

Solution:         Group the first two and the last two terms together, and factor out the common factor from each grouping:

(xy 4y) + (3x + 12) = y(x 4) + 3(−x + 4)

This time there is no common factor.  Try again, this time factoring a 3 from the last              grouping.  This works!

xy 4y   3x + 12 = y(x 4) 3(x 4)

= (x 4)(y 3)

EXAMPLE 4.             Factor  xy 4y + 3x + 12

Solution:             Group the first two and the last two terms:

xy 4y + 3x + 12 = y(x 4) + 3(x + 4)

At this point, it is important to realize that no common factor resulted.  Do not try to factor out something that is not common to both groupings.  In fact, there is no way to group this problem to get a common factor.  This one cannot be factored by grouping.  In fact, it can=t be factored by any method.  Remember, not all problems can be factored.  Remember that the entire process of grouping is one of trial and error, and, as you will see later, there are different types of grouping.

EXERCISES.             Factor each of the following by grouping:

1.         xy + 7x + 4y + 28                                            2.         2xy + 5x + 10y + 25

= x(          ) + 4(          )

= (           )(           )

3.          x3 + 3x2 + 9x + 27                                          4.         x3 3x2 + 9x 27

5.        ax + bx  +  ay  +  by                                        6.        ax +  ac + bx  + bc

7.        ax    bx  +  ay    by                                      8.        ax    ac +  bx    bc

9.        ax    bx    ay  +  by                                      10.       ax    ac    bx  +  bc

= x (a b)    y ( a b)                                            = ___(         ) ___(         )

= (             )(             )                                                = (               )(               )

11.       xy 5x 2y + 10                                           12.       x2y + xy2 5x 5y

=                                                                                 = xy(         ) 5(         )

=                                                                                 =

13.       x3 x2 9x + 9                                              14.       x3 5x2 4x + 20

= x2 (        ) 9(         )                                                = x2 (         ) 4(         )

= (          )(             )                                                     =

= (          )(        )(        )                                                =

15.       x3 + 7x2 x 7                                              16.      x3 5x2 + 25x 125

= x2 (         ) 1(          )                                             =

=                                                                                  =

=                                                                                  Does  x2 + 25 factor?

17.       x3 + 5x2 25x 125                                      18.       x3 5x2 25x + 125

19.       x3 4x2 + 9x 36                                          20.       x3 + 4x2 9x 36

21.       x3 4x2 9x + 36                                          22.       x3 9x2 4x + 36

23.       x3   8x2   x  +  8                                          24.       x3    8x2  +  4x    32

SUM AND DIFFERENCE OF CUBES

In this section, formulas and procedures will allow you to factor expressions in the form x3 y3 and also x 3 + y 3.  Recall from previous sections that  x2 y2 = (x y)(x + y) and that x2 + y2 cannot be factored.  Begin with the multiplication problems:

(x y)( x2 + xy + y2 )  =  x3 + x2y + xy2

− x2y xy2 y3

=      x3                 y3

(x + y)( x2 xy + y2 ) =  x3 x2y + xy2

+ x2y xy2 + y3

=       x3        +      y3 .

This derives the formulas, known as the "sum and difference of cubes formulas."

 SUM AND DIFFERENCE OF CUBES FORMULAS     x3  −  y3 = (x − y)( x2 + xy + y2)   x3  +  y3 = (x + y)( x2 − xy + y2)

Translated into words, this means the sum or difference of two cubes can be factored into the product of a binomial times a trinomial.  Begin by taking the cube root of the perfect cubes.

In the difference formula, the binomial is "the first minus the second."  Then use this binomial to build the trinomial that follows: take the "square of the first" plus the "product of the first and second" plus the "square of the second."

The sum formula is similar, in that the formula begins with the binomial that is "the first plus the second".  Then use this binomial to build the trinomial that follows, which except for one sign is the same as the difference formula: take the "square of the first" minus the "product of the first and second" plus the "square of the second."

These formulas are really easy to remember.  Notice that the   x3 y3 formula begins with (x y).   The x3 + y3  formula begins with (x + y).  Next, notice that the trinomial factor in both formulas is the same except for one sign.  This trinomial factor in each formula involves the "square of the first," the "product of the two," and the "square of the second."  The first sign in the trinomial is the opposite of the sign of the binomial, and the last sign is always positive.  Finally, you will never be able to factor the resulting trinomial (by ordinary trinomial methods), so you need not even try (at this level).

Before getting into the exercises, be sure to be familiar with the perfect cubes: 13 = 1; 23 = 8; 33 = 27; 43 = 64; and 53 = 125.  Be able to recite these from memory: 1, 8, 27, 64, 125.  Practice them as you drive down the highway!

EXERCISES.  Factor completely, using the sum and difference of cubes formulas.

25.       x3     8                                                   26.          x3    125

=  x3 23 [x=first; 2=second]                               = (       )3 (       )3

= (            )(x2  +  2x  +  22)                               = (        )(      +        +       )

= (             )(                       )                                 = (          )(                         )

27.       x3    64                                                  28.            x3    27

= (       )3 (       )3                                               = (       )3 (        )3

= (           )(                             )                             = (           )(                         )

29.       x3   +   8                                                  30.            x3   +   64

=  x3  +  23    [x=first; 2=second]                           = (       )3 + (       )3

= (      +     )(x2        +        )                                = (     +     )(              +        )

= (             )(                       )                                 = (            )(                            )

31.       x3  +  125                                                32.         x3 +  27

= (        )3 + (        )3                                              = (      )3 + (      )3

= (              )(                         )                              = (             )(                            )

33.       8x3    125                                              34.       27x3       8y3

= (2x)   53     [2x=first; 5=second]                   = (         )3 (         )3

= (            )[(2x)2 + (2x)(5) + 52]                        = (            )[       +       +       ]

= (              )(                            )                           = (             )(                          )

35.       64x3   +   125                                          36.          27x3   +   8y3

= (        )3 + (        )3                                               = (         )3 + (         )3

= (           )[                       ]                                    = (            )[                       ]

= (           )(                        )                                   = (            )(                        )

37.       8x3       27y3                                           38.         125y3     8x3

39.       8x3   +    1                                                  40.            125y 3        1

In the next exercises, don't forget to factor the common factor first.

41.       16x4    54x                                               42.            3x3       24y3

= 2x(8x3 27)                                                     =

= 2x[( 2x )3 ( 3 )3 ]                                            =

= 2x(___ ___)(____+____+____)                    =

43.       5x4 + 40x                                                   44.            10x5y  +  80x2y4

45.       3x5y5    81x2y2                                         46.            16x2y2 + 250x2y5

p. 174-180:      (NOTE:  Factors may be given in any order!)

1. (y+7)(x+4);  2. (2y+5)(x+5);  3. (x+3)(x2+9);  4. (x-3)(x2+9);  5. (a+b)(x+y);

6. (x+c)(a+b);    7. (a-b)(x+y);    8. (x-c)(a+b);    9. (a-b)(x-y);   10. (x-c)(a-b);

11. (y-5)(x-2);   12. (x+y)(xy-5);   13. (x-1)(x-3)(x+3);   14. (x-5)(x-2)(x+2);

15. (x+7)(x-1)(x+1); 16. (x-5)(x2+25);  17. (x+5)2(x-5); 18. (x-5)2(x+5); 19. (x-4)(x2+9);

20. (x+4(x-3)(x+3);   21. (x-4)(x-3)(x+3);  22. (x-9)(x-2)(x+2);  23. (x-8)(x-1)(x+1);

24. (x-8)(x2+4);   25. (x-2)(x2+2x+4);   26. (x-5)(x2+5x+25);   27. (x-4)(x2+4x+16);

28. (x-3)(x2+3x+9);   29. (x+2)(x2-2x+4);    30. (x+4)(x2-4x+16);   31. (x+5)(x2-5x+25);

32. (x+3)(x2-3x+9);   33. (2x-5)(4x2+10x+25);   34. (3x-2y)(9x2+6xy+4y 2);

35. (4x+5)(16x2-20x+25);  36. (3x+2y)(9x2-6xy+4y2);  37.  (2x-3y)(4x2+6xy+9y2);

38. (5y-2x)(25y2+10xy+4x2);  39.  (2x+1)(4x2-2x+1);  40.  (5y-1)(25y2+5y+1);

41. 2x(2x-3)(4x2+6x+9);  42. 3(x-2y) (x2+2xy+4y2); 43. 5x(x+2)(x2-2x+4);

44. 10x2y(x+2y)(x2-2xy+4y2); 45. 3x2y2(xy-3)(x2y2 +3xy+9); 46. 2x2y2(2+5y)(4-10y+25y2).

 Dr. Robert J. Rapalje Altamonte Springs Campus Contact me at: Phone number:  NONE Retired!! OFFICE:          NONE Copyright © Seminole State College of Florida, 1997

This lesson has been accessed 25020 times.