MA 15800
Lesson 4 Notes
Simplifying Algebraic Expression (part 2)
Summer 2016
polynomial
; a fraction of polynomial expressions.
polynomial
To simplify a rational expression means…
1.
The answer is in ‘lowest terms’; the numerator and denominator have no common
factors.
2.
Usually, the answer may be left factored.
A rational expression is of the form
Simplify each rational expression.
60
Ex 1:

220
Ex 3:
y 2  25

y 3  125
Ex 2:
48

12  4
Ex 4:
12  4r  r 2

r 3  2r 2
Simplify each expression using addition, subtraction, multiplication, or division.
5
3
5 7
Ex 5:
Ex 6:


 
24 20
12 8
Ex 7:
14 25
 
25 24
Ex 8: Divide:
5a 2  12a  4 25a 2  20a  4


a 4  16
a 2  2a
1
MA 15800
Lesson 4 Notes
Simplifying Algebraic Expression (part 2)
Summer 2016
For addition and subtraction (as noted earlier), you need to find a least common denominator
(LCD) and re-write both rational expressions with that LCD before adding or subtracting. Here
are a few examples of finding an LCD.
Find the least common denominator of each problem.
2
12
1
Ex 9:
 2
 2
2
n  3n  10 n  4n  4 n  5n
Ex 10:
4
x
 2
2
x  6 x  9 x 2 x  8x  6
3
Add or Subtract:
3m
5m
40
Ex 11:

 2

m2 m2 m 4
2
MA 15800
Lesson 4 Notes
Simplifying Algebraic Expression (part 2)
Ex 12:
4x
8
2
 2
 
3x  4 3x  4 x x
Ex 13:
2x  6
5x
7
 2


x  6x  9 x  9 x  3
Summer 2016
2
3
MA 15800
Lesson 4 Notes
Simplifying Algebraic Expression (part 2)
Summer 2016
Many mathematicians do not consider an expression left with a radical in its denominator to be
simplified. Clearing a radical from a denominator is called rationalizing the denominator.
Examine the following example.
Ex 14:
2
2
7


7
7 7
Ex 15:

2 7
49

2 7
2
or
7
7
7
Rationalize the denominator:
5 3

28
(It is easier to rationalize the denominator if the radicals are simplified first.)
Ex 16:
Rationalize the denominator. Assume the variable is positive.
2
18x3

4
MA 15800
Ex 17:
Lesson 4 Notes
Simplifying Algebraic Expression (part 2)
Rationalize the denominator.
Summer 2016
4

3
9
Conjugates are expressions of the form a  b and a  b . When rationalizing expressions with a
sum or difference in the denominator (with square roots), the numerator and denominator must
be multiplied by the conjugate of the denominator. Why do we do this? Because the product of
conjugates is always a rational number or expression (no roots). Examine the following.
(2  11)(2  11)  4  2 11  2 11  121
 4  121
 4  11
 7
 4  3 x  4  3 x   16  12
x  12 x  9 x 2
 16  9 x 2
 16  9 x
( 3  6)( 3  6)  9  18  18  36
 36
 3
When multiplying conjugates with square roots, the square root is always eliminated!
When rationalizing a denominator with a binomial denominator with a square root, multiply
numerator and denominator by the conjugate of the denominator. See the example 18 below.
Ex 18:
t  5 ( t  5) ( t  5)


t  5 ( t  5) ( t  5)


t 2  5 t  5 t  25
t 2  25
In the denominator, the 'inner' and 'outer' products were eliminated.
t  10 t  25
t  25
5
MA 15800
Lesson 4 Notes
Simplifying Algebraic Expression (part 2)
Ex 19:
Rationalize the denominator.
4 3

4 3
Ex 20:
Rationalize the denominator and simplify.
81x 2  16

3 x 2
Summer 2016
Occasionally in calculus there is a need to rationalize the numerator.
Ex 21: Rationalize the numerator in this expression. Simplify.
2 x 3

4 x2  3
6
MA 15800
Lesson 4 Notes
Simplifying Algebraic Expression (part 2)
Summer 2016
A complex fraction or complex rational expression is a quotient in which the numerator
and/or the denominator is(are) a fractional expression or expressions.
To simplify a complex rational expression, combine the numerator and/or the denominator into a
single quotient (one fraction). Convert to a division problem and perform the division.
Ex 22: Simplify the following expression.
3
3
 2
2
( x  h)
x

h
Ex 23: Simplify the following.
7
7

xh2 x2 
h
7
MA 15800
Ex 24: Simplify:
Lesson 4 Notes
Simplifying Algebraic Expression (part 2)
Summer 2016
x
a

x3 a3 
xa
Ex 25: Simplify this expression.
3
3

2 x  1 2a  1 
xa
8
MA 15800
Lesson 4 Notes
Simplifying Algebraic Expression (part 2)
Ex 26: Simplify this complex expression.
Summer 2016
b a

a b 
1 1

a b
Ex 27: Simplify:
1
3
x2

4
x
x
Ex 28: Simplify:
x

y
x2

y2
y
x

y2
x2
9
MA 15800
Lesson 4 Notes
Simplifying Algebraic Expression (part 2)
Summer 2016
Ex 29: Simplify:
5

x 1
x

x 1
Ex 30: Simplify:
2x
x3 
7
x3
( x  h)3  5( x  h)  ( x3  5 x)

h
Express each as a quotient. (Examples 10 – 11)
Ex 31: x 4  x 
Ex 32: x 3  x 
10
MA 15800
Lesson 4 Notes
Simplifying Algebraic Expression (part 2)
Summer 2016
For this next type of example, usually the answer should be left in factored form.
Ex 33: x

2
3
7
 x3 
These next types of examples will occur often in calculus when finding some types of
derivatives.
Ex 34: Factor the expression below.
(6 x  5)3 (2)( x2  4)(2 x)  ( x2  4)2 (3)(6 x  5)2 (6) 
Ex 35: Factor the expression below.
4 x2 ( x  1)2 ( x  2)  3x( x  1)( x  2)2 
11
MA 15800
Lesson 4 Notes
Simplifying Algebraic Expression (part 2)
Summer 2016
Ex 36: Factor the expression below.
( x 2  2)3 (2 x)  x 2 (3)( x 2  2) 2 (2 x)

2
( x 2  2)3 
Ex 37: Express as a quotient: x3/2  x1/2 
Ex 38: Simplify the expression.
(6 x  5)3 (2)( x2  4)(2 x)  ( x2  4)2 (3)(6 x  5)2 (6) 
12