FACTORING EXPRESSIONS
 

Objective:  To learn how to use the distributive law in reverse, a useful tool that will become handy when factoring is important.

Recall 1: You learned how multiplication distributes itself over addition as in

x * (y + z) = x * y + x * z.

It is often useful to apply this rule in reverse, as in

x * y + x * z = x * (y + z).

Recall 2: The process of applying the distributive law of multiplication over addition in reverse is usually referred  to as factoring out the common factor or just plain factoring.
The expression x*y + x*z has a common factor of x in both terms. Thus, to factor the expression x*y+x*z means to change it into the equivalent expression x*(y + z).

Examples

Example 1. Factor the expression 2*x + 8.

Solution: The expression 2*x + 8 may be rewritten as 2*x + 2*4. If we now apply the distributive law in reverse, then we observe that

2*x + 8 = 2*(x + 4).

Example 2. Factor the expression 5*x*x + 25*x*y.

Solution: The expression 5*x*x + 25*x*y may be rewritten as 5*x*x +  5*5*x*y.  Apply the commutative law for multiplication to this last  expression to make it become

5*x*x + 5*x*5*y.

 Next, apply the distributive law in reverse to arrive at

5*x*(x + 5*y).

Example 3. Factor the expression 2*t*r*h + 2*t*r*r.

Solution: The expression  2*t*r*h + 2*t*r*r may be written, applying the distributive law in reverse to factor out the 2*t*r, as

2*t*r*(h + r)

Example 4 Factor the expression (5*x)*(x + 3*y) - (2*y)*(x + 3*y).

Solution: Recall that - (2*y) is the same as + (-2)*y.  Therefore the  expression may be written as

(5*x)*(x + 3*y) + ((-2)*y)*(x + 3*y).

 By the commutative law for multiplication (in each term), the last  expression may be written as

(x + 3*y)*(5*x) + (x + 3*y)*((-2)*y).

 Notice that each of the terms of the expression above has the same  factor, namely (x + 3*y).  So, using the distributive law for  multiplication over addition we may factor out the common term (x + 3*y) to get
(x + 3*y)*(5*x + (-2*y)),

which, in turn, may be rewritten as

(x + 3*y)*(5*x +(- 2)*y).

Example 5. Factor the expression 5*x^2 + 15*y*x - 2*x*y - 6*y^2.

Solution: Applying the distributive rule to the first two terms and the  last two terms of the expression

5*x^2 + 15*y*x - 2*x*y - 6*y^2 gives

(5*x)*(x + 3*y) - (2*y)*(x + 3*y).

 Do you see how to proceed from here?  If not, take another look at  Example 1.

Reminder 1:    Remember to check your results.  For the  examples above, such checking is easy--for each example, pick x = 1 and evaluate the beginning expression and the end expression. When  you plug x = 1 into the beginning expression of Example 1, you find  that

2*(1) + 8 = 10,

 and if you plug x = 1 into the end expression, you find that

2*(1 + 4) = 10

 Therefore, you should expect to be correct.
Note:  It is possible to be  incorrect, even though you checked your result at x = 1.

EXERCISES
 

Factor the following expressions:

1. 5*x + 25

2. 3*x*x + 18

3  6*x^2 + 24*x

4. 2*(x + 4)^2 - 3*(x + 4)          [Hint: write (x + 4)^2 as (x + 4)*(x + 4).]

5. 1.43*x*y + 2.86*y^2

6. 2*y*(x - 2) + 6*y^2*(x - 2)

7. 2*z*x  - 3*x*y + 4*x^2*y^2 - 3*y*x

8. 3*a*(b + 3)^2 + 2*b*a*(b + 3) - a*(b + 3)            [Hint: write (b + 3)^2 as   (b + 3)*(b + 3).]

9. 4*a*b^2*c  - 2*b^2*c + 3*(2*a - 1)*c    [Hint: factor 2* b^2*c from  the first two terms.]

10. 2*y*(x + 2) + 6*y^2*(2 + x) [Hint: write (2 + x) as (x + 2).]

11. 2*x (5 - z) - 3*y*(z - 5)     [Hint: write (z - 5) as -(5 - z).]

12. x^2 - x*y - y^2 + x*y [Hint:Factor the x from the first two terms, and the y from the second two terms.]

13. 6*x^2 - 2*x*y + 9*x*y - 3*y^2      [Hint: Factor 2*x from the first two terms and 3*y from the last two terms.]

14. 2*A*B*h + 3*h*A*t

15. 4*a*((c + b)^2 + 3*(c + b))

16. 7*((x^2 - 2*x + 1) - (x - 1))                 [Hint:  x^2 - 2*x + 1 = (x - 1)^2.]

Checking for common errors:

1. You may have made some common errors in the last workout. So it is  advisable to test your answers by evaluating the expressions at x = 1, as  suggested in Reminder 1, above.

2. If you find yourself making many errors, slow down and carefully look  for the common factors in the terms of the expression.

3. Do not be too hasty to multiply.  In an exercise such as Exercise 4, the  expression is 2*(x + 4)^2 - 3*(x + 4).  There is a tendency to multiply out  the second term to get 2*(x + 4)^2 - 3*x + 12.  Then you will be forced to  multiply out the (x + 4)^2 and distribute the 2 to get 2*x^2 + 16*x + 32 -  3*x + 12.  Although it is possible to arrive at the correct answer in this  way, it is a long winded method.  You would do better to notice that in  the original expression 2*(x + 4)^2 - 3*(x + 4) there are two terms, each  containing the factor (x + 4).  So, the expression may be rewritten as  2*(x + 4)*(x + 4) - 3*(x + 4), which permits you to factor out a copy of  (x + 4) to get

(x + 4)*(2*(x + 4) - 3),

 which may be simplified to

(x + 4)*(2*x + 8 - 3),
 and then to
(x + 4)*(2*x + 5).

 In general, find common factors, before multiplying out terms. Remember that a common factor does not necessarily have to look like a  single symbol, but can be an entire parenthetical expression itself.

 Can you find the errors in the solutions to Exercises  below?  (Note: these are samples from students who volunteered to  work through the first draft of Algebra Workout.)

Incorrect solution
to Exercise 8
                                          3*a*(b + 3)^2 + 2*b*(b + 3) - (b + 3),
 Using the hint ->                 3*a*(b + 3)*(b + 3) + 2 b*(b + 3) - (b + 3),
Factoring out (b + 3) ->        (b + 3)*(3*a*(b + 3) + 2*b),
                                          (b + 3)*(3*a*b + 9*a + 2*b).
 

Incorrect solution
to Exercise 9
                                             4*a*b2*c  - 2*b^2*c + 3*(2*a - 1)*c,
 Using the hint ->                     2 b^2*c*(2*a - 1) + 3*(2*a - 1)*c,
 Factoring out (2*a - 1) ->         2*b^2*c*(2*a - 1)*(1 + 3*c).
 

Incorrect solution
to Exercise 10
                                                               2*y*(x + 2) + 6*y^2*(2 + x),
Using the commutative law on (2 + x) ->     2*y*(x + 2) + 6*y^2*(x + 2),
Factoring out the (x + 2) ->                       (x + 2)*(2*y + 6*y^2),
                                                               (x + 2)*(8*y).
 

Incorrect solution
to Exercise 11
                                                   2*x (5 - z) - 3*y*(z - 5),
 Commutative law (misused) ->       2*x*(5 - z) - 3*y*(5 - z),
                                                     (5 - z)*(2*x  - 3*y).
 

Incorrect solution
to Exercise 12
                                                           x2 - x*y - y^2 + x*y,
 Using the hint with a sign mistake ->      x*(x - y) - y*(y + x),
 Commutative law and sign error ->     x*(x - y) + y*(x - y),
                                                         (x - y)*(x + y).
 

What if you didn't use
the hint in Exercise 13?

        6*x^2 - 2*x*y + 9*x*y - 3*y^2,
 Ignoring the hint ->  6*x^2 + 7*x*y - 3*y^2,
       **** Stuck! ****

 If you ignored the hint, then it is likely that you got stuck at this point.  That's because  ignoring the hint makes the problem harder.  Now you must guess at how the expression factors.  It turns out that the expression factors as
   (2*x + 3*y)*(3*x - y).
 
 

ANSWERS

1. 5*(x + 5)

2. 3*(x^2 + 6)

3.  6*x*(x + 4)

4. (x + 4)*(2*x + 5)

5. 1.43*y*(x + 2*y)

6. 2*y*(x - 2)*(1 + 3*y)

7. 2*x*(z - 3y + 2*x*y^2)

8. a*(b + 3)*(5*b + 8)

 9. c*(2*a - 1)*(2*b^2 + 3)

10. 2*y*(x + 2)*(1 + 3*y)

11. (5 - z)*(2*x + 3*y)

12. (x - y)*(x + y)

13. (2*x + 3*y)*(3*x - y)

14. A*h*(2*B + 3*t)

15. 4*a*(c + b)*(c + b + 3).

16. 7*(x - 1)*(x - 2).