Download M098 Carson Elementary and Intermediate Algebra 3e Section 7.1 Objectives

M098
Carson Elementary and Intermediate Algebra 3e
Section 7.1
Objectives
1.
2.
3.
4.
Evaluate rational expressions
Find numbers that cause a rational expression to be undefined.
Simplify rational expressions containing only monomials.
Simplify rational expressions containing multiterm polynomials.
Vocabulary
Rational expression
An expression that can be written in the form
P
, where P and Q are polynomials and
Q
Q ≠ 0.
Prior Knowledge
Rational numbers are numbers that can be expressed in the form a/b where a and b are integers and b≠0.
Fractions are undefined if the denominator equals 0.
To evaluate means to replace the variable(s) with the given value(s).
New Concepts
1. Evaluate rational expressions.
This chapter deals with rational expressions and rational equations. The word association for rational is
fraction. We’ll spend some time reviewing our basic fraction arithmetic skills. We use exactly the same
process with rational expressions (algebraic fractions) as we do with arithmetic fractions - the problems
are just a little uglier looking!!!
To evaluate a rational expression, replace the variable(s) with the given values.
Example 1: Evaluate
3(2)  2
4(2)  5
3x  2
when x = 2.
4x  5
Replace the variable with the given number.
62
85
8
3
Example 2: Evaluate
4(3)
(3 )  3
12
0
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Evaluate the numerator and the denominator.
4x
when x = 3.
x3
Replace the variable with the given number.
Evaluate the numerator and the denominator. This
expression is undefined.
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M098
Carson Elementary and Intermediate Algebra 3e
Section 7.1
2. Find numbers that cause a rational expression to be undefined.
One of the first concepts in Chapter 1 was that a fraction can not have 0 in the denominator. It is
undefined when that occurs. The numerator (top) can be 0 but not the denominator. The same is true in
this chapter. Since rational expressions have variables in the denominator, we have to continually remind
ourselves to check to see which numbers will make the fraction undefined - in other words, when will the
denominator equal 0.
Example 3: When will
x2
be undefined?
x5
This will be undefined when the denominator (x – 5) equals 0.
x–5=0
Solve the linear equation.
x=5
The fraction will have 0 in the denominator if x = 5.
The rational expression is undefined when x = 5.
Example 4: When will
x2  4
x 2  7x  10
be undefined?
2
This will be undefined when the denominator (x – 7x + 10) equals 0.
2
x - 7x + 10 = 0
Solve the quadratic equation. There should be 2 solutions.
(x - 2) (x - 5) = 0
x-2=0
x-5=0
x=2
x=5
The fraction will have 0 in the denominator if x = 2 or x = 5.
The rational expression is undefined when x = 2 or x = 5.
Example 5: When will
2p  3
p 2  6p
be undefined?
2
This will be undefined when the denominator (p + 6p) equals 0.
2
p + 6p = 0
Solve the quadratic equation. There should be 2 solutions.
p(p + 6) = 0
p=0 p+6=0
p = -6
The fraction will have 0 in the denominator if p = 0 or p = -6.
The rational expression is undefined when p = 0 or p = -6.
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M098
Carson Elementary and Intermediate Algebra 3e
Section 7.1
3. Simplify rational expressions.
In working with fractions in arithmetic, we always reduced them to lowest terms by dividing the numerator
and denominator by the greatest common factor. If you remember back to when you first learned to
reduce fractions, the process went something like this:
18
233
3


24
234
4
We can reduce these fractions because 2 and 3 are common factors. This is another one of those red
flag areas. We can only reduce when we are multiplying - never when we add.
23
5
3
which does not equal .

25
7
5
Terms can NOT be reduced – only factors.
Example 6: Simplify
14h 3k
21 h
27hhhk
37h
Rewrite in factored form.
27hhhk
37h
Reduce common factors.
2h 2k
3
Multiply remaining factors.
Example 7: Simplify
x 2  10x  25
x 2  25
.
x  5x  5
x  5x  5
Rewrite in factored form.
x  5x  5
x  5x  5
Reduce common factors.
x5
x5
Example 8: Simplify
by  ay  bx  ax
x 2  ax  xy  ay
.
b  ay  x 
x  ax  y 
Rewrite in factored form.
b  ay  x 
x  ax  y 
Reduce common factors.
ba
xa
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M098
Carson Elementary and Intermediate Algebra 3e
Example 9: Simplify
Section 7.1
x 3  64
.
4x
x  4x 2  4x  16
Rewrite in factored form.
4x
x  4x 2  4x  16
 x  4 
x  4x 2  4x  16
 x  4 
x 2  4x  16
1
 1 x 2  4x  16


(x – 4) and 4 – x are inverses. Factor out -1 so from one
of them so the binomials match.
Reduce common factors.
It is common practice to remove the minus from the
denominator. The -1 can be moved to the numerator.
Distribute the -1.
 x 2  4x  16
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