lesson 8-1: multiplying and dividing rational expressions
TRANSCRIPT
Lesson 8-1: Multiplying and Dividing Rational Expressions
Rational Expression
Definition: a ratio of two polynomial expressions
x
x
13
8
To Simplify A Rational Expression
1. Make sure both the numerator and denominator are factored completely!!!
2. Look for common factors and cancel– Remember factors are things that are being
multiplied you can NEVER cancel things that are being added or subtracted!!!
3. Find out what conditions make the expression undefined and state them.
Examples: Simplify and state the values for x that result
in the expression being undefined
)128(6
632
yyy
yy 6412
8742
23
xx
xxx2.1.
Examples Cont… Simplify
wzz
zwz33
22
yxx
xxy223
3
baa
aba33
44
2
2
5.4.3.
Operations with Rational Expressions
To Multiply Rational Expressions:
Factor and cancel where possible. Then multiply numerators and denominators
To Divide Rational Expressions:
Rewrite the problem as a multiplication problem with the first expression times the reciprocal of the second expression. Then factor and cancel where possible. Multiply numerators and denominators
Define x-values for which the expression is undefined
Examples: Simplify
xy
b
yx
ab
10
9
25
1832
2
3
2
16
15
5
4
a
b
a
a
232
2
12
15
5
8
st
sr
r
st
83
3
37
4
35
21
15
14
zw
qp
zw
pq
7.
9.8.
6.
Polynomials in Numerator and Denominator
Rules are the same as before…1. Make sure everything is factored completely
2. Cancel common factors
3. Simplify and define x values for which the expression is undefined.
Examples: Simplify and define x values for which it is undefined
9
12
3
22
2
a
aa
a
a54
65
155
12
2
yy
yy
y
y10.11.
Examples:
34
1
1
32
2
kk
k
k
k23
3
2
6222
dd
d
dd
d13.12.
Simplifying complex fractions
A complex fraction is a rational expression whose numerator and/or denominator contains a rational expression
b
a
35
3
3
tt
t
tt
6
3 2
To simplify complex fractions
Same rules as before– Rewrite as multiplication with the reciprocal– Factor and cancel what you can– Simplify everything– Multiply to finish
Examples:
43163
2
2
xxxx14.
xyxyx
x
32
493
22
2
15.
Lesson 8-2: Adding and Subtracting Rational Expressions
Adding and Subtracting Rational Expressions
Finding Least Common Multiples and Least Common Denominators!– Use prime factorization
– Example: Find the LCM of 6 and 4 6 = 2·3 4 = 22
LCM= 22·3 = 12
Find the LCM
1. 18r2s5, 24r3st2, and 15s3t
2. 15a2bc3, 16b5c2, 20a3c6
3. a2 – 6a + 9 and a2 + a -12
4. 2k3 – 5k2 – 12k and k3 – 8k2 +16k
Add and Subtract Rational Expressions
Same as fractions…– To add two fractions we find the LCD, the same
things is going to happen with rational expressions
Examples: Simplify
27
1
9
8
abb
a
xy
y
y
x
1815
72
22
1
8
1
mnnm
5.
7.
6.
82
4
164
12
w
w
w
w
62
6
186
6
x
x
x
x
2012
1
583
12
x
x
xx
x8. 9. 10.
xy
xy11
11
x
yx1
1
11
11. 12.
xx
x4
42
14.
6
3x
x
xx
13.
Lesson 8-3: Graphing Rational Functions
Definitions
Continuity: graph may not be able to be traced without picking up pencil
Asymptote: a like that the graph of the function approaches, but never touches (this line is graphed as a dotted line)
Point discontinuity: a hole in the graph
Vertical Asymptote
How to find a Vertical Asymptote:
x = the value that makes the rational expression undefined
*Set the denominator of the rational expression equal to zero and solve.
Point Discontinuity
How to find point discontinuity:
* Factor completely
* Set any factor that cancels equal to zero and solve. Those are the
x values that are points of discontinuity
Graphing Rational Functions
f(x) = 1
3
x
f(x) =
Graphing Rational Functions
x
2
f(x) =
Graphing Rational Functions
3
12
x
x
f(x)=
Graphing Rational Functions
2)4(
2
xx
f(x) =
Graphing Rational Functions
3
92
x
x
f(x) =
Graphing Rational Functions
1
562
x
xx
Lesson 8-4: Direct, Joint, and Inverse Variation
Direct Variation
y varies directly as x if there is a nonzero constant, k, such that y = kx*k is called the constant of variation
1. Plug in the two values you have and solve for the missing variable
2. Plug in that variable and the other given value to solve for the requested answer
Example
If y varies directly as x and y = 12 when
x = -3, find y when x = 16.
Joint Variation
y varies jointly as x and z if there is a nonzero constant, k, such that y = kxz
* Follow the same directions as before
Example
Suppose y varies jointly as x and z. Find y when x = 8 and z = 3, if y = 16 when z = 2 and x = 5.
Inverse Variation
y varies inversely as x if there is a nonzero constant, k, such that xy = k or y=
kx
Example
If y varies inversely as x and y = 18 when x = -3, find y when x = -11
Lesson 8-6: Solving Rational Equations and Inequalities
Let’s review some old skills
How do you find the LCM of two monomials– 8x2y3 and 18x5
* Why do we find LCM’s with rational expressions?
Old Skills Cont…
What is it called when two fractions are equal to each other?
What process do we use to solve a problem like this?
To solve a rational equation:
1. Make sure the problem is written as a proportion
2. Cross Multiply
3. Solve for x
4. Check our answer
Examples
Solve
9
1
52
1x
102
3
3
2
x
Let’s put those old skills to new use…
Solve .
Solve . Check your solution.
3
3
158
10
5
22
2
xxx
xx
x
x
Examples
Solve
22
3
44
5
33
7
n
n
nn
n
Lesson 8-6: Day #2
Rational Inequalities
Solving Rational Inequalities
Step 1: State any excluded values (where the denominator of any fraction could equal zero)
Step 2: Solve the related equation
Step 3: Divide a numberline into intervals using answers from steps 1 and 2 to create intervals
Step 4: Test a value in each interval to determine which values satisfy the inequality
Examples
Solve
4
1
2
1
3
x
x
x
Examples
Solve
3
2
5
65xx
Examples
Solve
2
4
2
1
2
2
x
x
xx
x
Examples
Solve
2
3
1
1
2
xx
x