Download Chapter R.5 Introduction to Rational Expressions

Chapter R.5 Introduction to Rational Expressions
A rational expression is a ratio of polynomials; that is, a fraction that has a
polynomial as a numerator and/or denominator.
Sample rational expressions:
x 2 + 3 x − 10
4x 3 − 1
3x
x+5
5
x −1
● EVALUATING RATIONAL EXPRESSIONS
To evaluate a rational expression for a given value of x, simply substitute the value for
x and simplify.
− 2x 2 + 6x
if x = −2
x+4
− 2(−2) 2 + 6(−2) − 2(4) − 12 − 8 − 12 − 20
= −10
=
=
=
(−2) + 4
2
2
2
Sample Problem:
Solution:
Evaluate
Student Practice: Evaluate the following rational expressions if x = 3.
x2 −1
1.
3x − 5
x 3 − 25
2.
x−3
● DETERMINING WHEN A RATIONAL EXPRESSION IS UNDEFINED
A rational expression is undefined when the value of it’s denominator is 0 because
dividing by 0 is not allowed. To determine for what value of x a rational expression is
undefined, simply set the denominator equal to 0 and solve for x.
− 3x 2 + 7
undefined.
4 x − 10
Solution: We need to determine what makes the denominator 0, so we start by:
4 x − 10 = 0
Check:
4 x = 10
5
x=
2
Sample Problem:
Determine what value of x makes
Student Practice:Determine what value of x makes each rational expression undefined.
3.
16
x−3
x5
5.
3x − 7
4.
3x 3 − 9 x 2
8x
6.
3x − 4
8 x − 40
● SIMPLIFYING RATIONAL EXPRESSIONS
Simplifying rational expressions is much like simplifying fractions. First factor each
polynomial, then cancel like factors. (Use caution when canceling)
Tip: You can NOT cancel any individual term that is being ADDED OR SUBTRACTED. You can
ONLY cancel factors, or terms that are being MULTIPLIED!
Sample Problem: Simplify.
Solution:
x 2 − 16
x 2 + 5 x − 36
( x − 4)( x + 4) ( x − 4)( x + 4) x + 4
x 2 − 16
=
=
=
2
x + 5 x − 36 ( x + 9)( x − 4) ( x + 9)( x − 4) x + 9
Student Practice: Simplify each rational expression.
16 y
9x 2
7.
8.
20 y 3
36 x
9.
11.
4 x − 12
8x
4x + 8
2
x − 3 x − 10
10.
x 2 − 25
x+5
12.
2x 2 + 6x
3x 2 + 9 x
Review on Radicals
● Introduction to Radical Expressions
The symbol
is called the square root and is defined as follows:
a = c only if
Sample Problem:
Solution:
Simplify
16 = 4
c2 = a
16
since
4 2 = 16 .
Note that every positive number has two square roots, a positive and a negative root.
For example, the square roots of 16 are 4 and -4, since 4 2 = 16 and (−4) 2 = 16 . The
symbol implies the positive root, or the principal square root. To get the negative
root, a negative sign must be used in front of the square root sign as in −
Sample Problem:
Simplify
Solution:
− 16 = −4
− 16
Student Practice: Simplify each of the following radical expressions.
1.
25
2. − 81
3.
1
4. − 36
5.
100
6. − 49
7.
− 9 (not real #.)
8.
y2
.
9.
x6
11.
x 16
13.
9 x 12
10.
w10
12.
16a 2 b 8
Tip: When taking roots of exponential expressions, keep the base and take half of the exponent.
● SIMPLIFYING RADICAL EXPRESSIONS
A radical expression is simplified when there are no perfect square factors inside the
radical; i.e., when you take as much as you can out of the radical. Simplifying can be
done using the following rule:
PRODUCT RULE
a•b = a • b
a • b = a•b
OR
The product rule can be used to simplify radicands that are not perfect squares.
Simply factor the radicand using a perfect square as a factor.
Sample Problem 1:
Simplify
36
36 = 4 • 9 = 4 • 9 = 2 • 3 = 6
Solution:
Sample Problem 2:
Simplify
45
Solution: Since 45 is not a perfect square, we first factor 45 using a perfect square
factor. It may help to list the first few perfect square factors, which are 1,4,9,16: In our
45 = 9 • 5 = 9 • 5 = 3 • 5 = 3 5
case, we’ll use 9.
______________________________________________________________________
FOR LARGER NUMBERS. For larger numbers, it can be helpful to factor the
radicand into prime factors. To do so, divide by the smallest prime number as man times
as possible, then the next smallest, then the next until you are left with a prime number.
For each pair of identical factors, .
5
Sample Problem 2: Simplify
240
3 15
2
2
2
2
30
60
120
240
Solution:
240 = 2 • 2 • 2 • 2 • 3 • 5 = 4 • 4 • 3 • 5 = 4 • 4 • 3 • 5 = 2 • 2 • 15 = 4 15
2
2
240 = 2 • 2 • 15 = 4 15
Tip: To check your answer, multiply the outer number twice and then multiply by the
inner number to get what you started with. 4 • 4 • 5 = 240
Sample Problem 3:
Solution:
Simplify
x 13
x 13 = x 12 • x = x 12 • x = x 6 x
Student Practice: Simplify each radical expression.
1.
18
2.
50
3.
27
4.
24
5.
32
6.
72
7.
21
8.
64
9. w 8
10.
x9
11.
a 15
12.
a3
13.
a 5 b10
14.
20 x 7