Transcript RATIONAL EXPRESSIONS

RATIONAL
EXPRESSIONS
Chapter 5
5-1 Quotients of
Monomials
Multiplication Rule for Fractions
Let p, q, r, and s be
real numbers with
q ≠ 0 and s ≠ 0.
Then
p •r = pr
q s qs
Rule for Simplifying Fractions
Let p, q,and r be
real numbers with
q ≠ 0. Then
pr = p
qr q
EXAMPLES
Simplify:
30
40
Find the GCF of the
numerator and
denominator.
EXAMPLES
GCF = 10
30 = 10 • 3
40
10 • 4
= 3/4
EXAMPLES
Simplify:
9xy3
2
2
15x y
Find the GCF of the
numerator and
denominator.
EXAMPLES
2
3xy
GCF =
3
2
9x = 3y • 3xy
15x2y2 5x • 3xy2
= 3y/5x
LAWS of EXPONENTS
Let m and n be
positive integers
and a and b be
real numbers,
with a ≠ 0 and b≠0
when they are
divisors. Then:
Product of Powers
•
m
a •
n
a
•
3
x
5
x
•
=
=
m
+
n
a
8
x
• (3n2)(4n4) = 12n6
Power of a Power
• (am)n = amn
•
3)5
(x
=
15
x
Power of a Product
•
m
(ab)
=
•
2
3
(3n ) =
m
m
a b
3
6
3n
Quotient of Powers
If m > n, then
am  an = am-n
5
x

2
x
=
3
x
6
4
(22n )/(2n )
=
2
11n
Quotient of Powers
If m < n, then
m
n
|m-n|
a  a = 1/a
5
x

7
x
=
2
1/x
3
9
(22n )/(2n )
= 11
n6
QUOTIENTS of MONOMIALS are SIMPLIFIED
When:
• The integral coefficients
are prime (no common
factor except 1 and -1);
• Each base appears only
once; and
• There are no “powers of
powers”
EXAMPLES
Simplify
4
3
24s t
32s5
EXAMPLES
Answer
3
3t
4s
EXAMPLES
Simplify
3
-12p q
4p2q2
EXAMPLES
Answer
-3p
q
5-2 Zero and Negative
Exponents
Definitions
If n is a positive
integer and a ≠ 0
0
a =1
a-n = 1
an
00 is not defined
Definitions
If n is a positive
integer and a ≠ 0
-n
a
=
n
1/a
EXAMPLES
Simplify:
-1
0
-3
-2 a b
EXAMPLES
Answer:
-1
2b3
5-3 Scientific Notation
and Significant Digits
Definitions
In scientific notation,
a number is
expressed in the
form m x 10n where
1≤ m < 10 and n is an
integer
5-4 Rational Algebraic
Expressions
Definitions
Rational Expression –
is one that can be
expressed as a
quotient of
polynomials, and is
in simplified form
when its GCF is 1.
EXAMPLES
Simplify:
x2 - 2x
x2 – 4
Factor
EXAMPLES
Answer:
x (x – 2)
(x + 2)(x - 2)
=x
x+2
Definitions
A function that is
defined by a
simplified rational
expression in one
variable is called
a rational function.
EXAMPLES
Find the domain of the
function and its zeros.
f(t) = t2 – 9
t2 – 9t
FACTOR
Answer:
EXAMPLES
(t + 3) (t – 3)
t(t – 9)
Domain of t = {Real
numbers except 0 and
9}
Zeros are 3 and -3
Graphing Rational Functions
1. Plot points for (x,y) for
the rational function.
2. Determine the
asymptotes for the
function.
3. Graph the asymptotes
using dashed lines.
Connect the points
using a smooth curve.
Definitions
Asymptotes- lines
approached by
rational functions
without intersecting
those lines.
5-5 Products and
Quotients of Rational
Expressions
Division Rule for Fractions
Let p, q, r , and s be
real numbers with q
≠ 0, r ≠ 0, and s ≠ 0.
Then p  r = p • s
q s q r
EXAMPLES
Simplify.
6xy  3y
a2
a3x
EXAMPLES
Answer:
= 6xy •a3x
2
a
3y
= 2ax2
5-6 Sums and
Differences of Rational
Expressions
Addition/Subtraction Rules for
Fractions
1. Find the LCD of the
fractions.
2. Express each fraction
as an equivalent
fraction with the LCD
and denominator
3. Add or subtract the
numerators and then
simplify the result
EXAMPLES
Simplify:
= _1 – _1 + _3_
6a2 2ab 8b2
4b2 – 12ab + 9a2
= 24a2b2
=
(2b – 3a)2
24a2b2
5-7 Complex Fractions
Definition
Complex fraction – a
fraction which has
one or more
fractions or powers
with negative
exponents in its
numerator or
denominator or both
Simplifying Complex Fractions
1. Simplify the
numerator and
denominator
separately; then
divide, or
Simplifying Complex Fractions
2. Multiply the
numerator and
denominator by the
LCD of all the
fractions appearing
in the numerator
and denominator.
EXAMPLES
Simplify:
(z – 1/z) (1 – 1/z)
Use Both Methods
5-8 Fractional
Coefficients
EXAMPLES
Solve:
x2 = 2x + 1
2 15 10
5-9 Fractional Equations
Definition
Fractional equation –
an equation in which
a variable occurs in
a denominator.
Definition
Extraneous root – a
root of the
transformed
equation but not a
root of the original
equation.
END