Rational Expressions
Martin-Gay, Developmental Mathematics 2
Rational Expressions
Examples of Rational Expressions
54
423 2
x
xx22 432
34
yxyx
yx
4
3 2x
Martin-Gay, Developmental Mathematics 3
To evaluate a rational expression for a particular value(s), substitute the replacement value(s) into the rational expression and simplify the result.
Evaluating Rational Expressions
Example
Evaluate the following expression for y = 2.
y
y
5
2)
2 2( 25
7
4
7
4
Martin-Gay, Developmental Mathematics 4
In the previous example, what would happen if we tried to evaluate the rational expression for y = 5?
y
y
5
2 5 25 5 0
3
This expression is undefined!
Evaluating Rational Expressions
Martin-Gay, Developmental Mathematics 5
We have to be able to determine when a rational expression is undefined.
A rational expression is undefined when the denominator is equal to zero.
The numerator being equal to zero is okay (the rational expression simply equals zero).
Undefined Rational Expressions
Martin-Gay, Developmental Mathematics 6
Find any real numbers that make the following rational expression undefined.
4515
49 3
x
xx
The expression is undefined when 15x + 45 = 0.
So the expression is undefined when x = 3.
Undefined Rational Expressions
Example
Simplifying Rational Expressions
Martin-Gay, Developmental Mathematics 8
Simplifying a rational expression means writing it in lowest terms or simplest form.
If P, Q, and R are polynomials, and Q and R are not 0,
Q
P
QR
PR
Simplifying Rational Expressions
Martin-Gay, Developmental Mathematics 9
Simplifying Rational Expressions
A “rational expression” is the quotient of two polynomials. (division)
2
2
nn
23
1032
x
xx
Martin-Gay, Developmental Mathematics 10
Simplifying Rational Expressions
A rational expression is in simplest form when the numerator and denominator have no common factors (other than 1)
15
9Simplify
53
33
5
3
Martin-Gay, Developmental Mathematics 11
Simplifying Rational Expressions
53
33Simplify
Martin-Gay, Developmental Mathematics 12
How to get a rational expression in simplest form…
Factor the numerator completely (factor out a common factor)
Factor the denominator completely (factor out a common factor)
Cancel out any common factors (not addends)
Martin-Gay, Developmental Mathematics 13
Difference between a factor and an addend
A factor is in between a multiplication sign
An addend is between an addition or subtraction sign
Example:
x + 3 3x + 9
x – 9 6x + 3
3( 3)
3(2 1)
x
x
3( 3)
3(2 1)
x
x
Martin-Gay, Developmental Mathematics 14
x
x
5
105 :Simplify
x
x
5
25
Factor
x
x 2
x
x 2
Martin-Gay, Developmental Mathematics 15
1
23 :Simplify
2
2
y
yy
11
21
yy
yy1
2
y
y
Martin-Gay, Developmental Mathematics 16
y
y
48
2412 :Simplify
yy
412
212
y
y
4
2
Martin-Gay, Developmental Mathematics 17
xx
xx
23
2 :Simplify
2
2
2 1
3 2
x x
x x
2 1
3 2
x
x
Martin-Gay, Developmental Mathematics 18
2
2
9Simplify:
2 15
a
a a
3 3
5 3
a a
a a
( 3)
( 5)
a
a
Martin-Gay, Developmental Mathematics 19
x
x
4
4 :Simplify
41
4
x
x
11
1
Martin-Gay, Developmental Mathematics 20
Wentz’s Shortcut
1a b
b a
Martin-Gay, Developmental Mathematics 21
Opportunity for Mathematical Growth
Please do your handout (17 questions)
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