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Rational Expressions

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Title: PowerPoint Presentation Author: Joyce DuVall Last modified by: PMSD Created Date: 10/10/2014 2:46:56 PM Document presentation format: On-screen Show (4:3) – PowerPoint PPT presentation

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Title: Rational Expressions


1
Rational Expressions
Student will be able to simplify rational
expressions And identify what values make the
expression Undefined .
a.16
2
Simplifying Rational Expressions
  • The objective is to be able to simplify a
    rational expression-These are already!

3
Undefined denominators
  • Ignore the numerator
  • Set the denominator to zero and solve

4
Undefined denominators-ex.
  • What value(s) would make these undefined

5
Undefined denominators-ex.
  • What value(s) would make these undefined

X20 x2 9 0 X-2 x 3 0 x 3
0 x -3 x 3
6
Try these
  • For what value of a are these undefined

7
Answers
  • 1. 4a 0
  • 4 4
  • a 0
  • 2. 3a2 0
  • -2 -2
  • 3a -2
  • 3 3
  • a -2/3

8
This is not reduced
We do not have to factor monomial terms.
9
The greatest common factor is 5divide it out
both parts.
10
Try these
11
Cancel all common factors.
Answers
12
Vocabulary
  • Polynomial The sum or difference of monomials.
  • Rational expression A fraction whose numerator
    and denominator are polynomials.
  • Domain of a rational expression the set of all
    real numbers except those for which the
    denominator is zero.
  • Reduced form a rational expression in which the
    numerator and denominator have no factors in
    common.

13
Simplifying Rational Expressions
  • Divide out the common factors
  • Factor the numerator and denominator and then
    divide the common factors

14
Dividing Out Common Factors
Step 1 Identify any factors which are common to
both the numerator and the denominator.
  • The numerator and denominator have a common
    factor.
  • The common factor is the five.

15
Dividing Out Common Factors
  • Step 2 Divide out the common factors.
  • The fives can be divided since 5/5 1
  • The x remains in the numerator.
  • The (x-7) remains in the denominator

16
Factoring the Numerator and Denominator
  • Factor the numerator.
  • Factor the denominator.
  • Divide out the common factors.
  • Write in simplified form.

17
Factoring
Step 1 Look for common factors to both terms in
the numerator.
  • 3 is a factor of both 3 and 9.
  • X is a factor of both x2 and x.

Step 2 Factor the numerator.

3
3
x
x
(
)
3
12
x
18
Factoring
Step 3 Look for common factors to the terms in
the denominator and factor.
  • The denominator only has one term. The 12
    and x3 can be factored.
  • The 12 can be factored into 3 x 4.
  • The x3 can be written as x x2.


3
3
x
x
(
)
2



3
4
x
x
19
Divide and Simplify
Step 4 Divide out the common factors. In this
case, the common factors divide to become 1.
Step 5 Write in simplified form.

x
3
2
x
4
20
You Try It
Simplify the following rational expressions.
21
Problem 1
Divide out the common factors.
Write in simplified form.
22
Problem 2
Factor the numerator and denominator
Divide out the common factors.
Write in simplified form.
23
Problem 3
Factor the numerator and denominator
Divide out the common factors.
Write in simplified form.
24
Problem 4
Factor the numerator and denominator
Divide out the common factors.
Write in simplified form.
25
Reducing to -1
Reduce
26
Answer
-1
27
Student will be able to Multiply Rational
Expressions and express in simplest form a2.a.16
Do Now Multiply
Copy this
28
Student will be able to Multiply Rational
Expressions and express in simplest form a2.a.16
Cross cancel common factors and then multiply
across The numerators and across the
denominators
9
4
29
Multiplying when factoring is necessary!
FACTOR
30
Canceling step
Cancel top and bottom and on diagonals
2
Multiply numerators, multiply denominators

31
Ex
32
Restrictions on Rational Expressions
It is undefined for any value of x which makes
the denominator zero.
The restriction is that x cannot equal 5.
33
YOU TRY IT
What are the excluded values of the variables for
the following rational expressions?
34
Problem 1
Solution y ? 0 z ? 0
35
Problem 2
Solution 2x - 12 0 2x - 12 12 0
12 2x 12 2x ? 2 12
? 2 x 6
ANSWER x ? 6
36
More complicated
What are the excluded values of the variables for
the following rational expression. ? (undefined)
37
Problem 3
Solution C2 2C - 8 0 (C-2)(C4) 0 C-2
0 or C 4 0 C 2
or C -4
Answer C ? 2 C ? -4
38
Dividing Rationals
Student will be able to divide rational
expressions and Express answer in equivalent
simplest form. Do Now divide these fractions
(remember that dividing is Multiplying by the
reciprocal)
39
Answer
Multiply by the reciprocal a.k.a. Flip and
multiply
2
1
40
Algebraic Example
Note after inverting, (flipping) the second
expression, factor all four parts and follow
multiplying rules
41
Algebraic Example
2
42
Example 2
(Completely factor the First numerator)
43
Example 2
(Completely factor the First numerator)
2
44
When division looks different
45
Example when one is not a fraction..
46
Do and hand in on exit card
47
Adding/Subtracting Rational Expressions
Do now (remember common denominators)
Today, you will be able to add rational
expressions by finding Least common
denominators..
48
Adding/Subtracting Rational Expressions
49
Algebraic examples
50
Algebraic examples
Lcd 6
Lcd6
51
Answers
Distribute!
52
Subtracting-remember to distribute!
53
Subtracting-remember to distribute!
But this can be reduced!
54
Reducing
55
Trickier denominators
Here we should factor the second denominator in
order to find The least common denominator
56
Finding the lcd
Which means (x3)(x 3) is the lcd so multiply
the first Fraction by (x 3)/(x 3)
57
Answer
Not reducable!
58
Next example
59
Solution
60
Try this-(factor to find lcd)
This one will need to be reduced at the end.
61
Answer lcd (x-5)(x3)
62
Answer
63
Complex Fractions a.17
Student will be able to simplify complex
fractions by Multiplying each term by the least
common denominator and Simpifying if necessary.
Do Now - Divide
64
A fraction over another fraction
Now think of it this way This is called a
complex fraction.
We flip the bottom and multiply, just Like when
we divided.
65
Fractions within a fraction
Step 1-find the lcd of all 4 terms Step
2-multiply each term by the lcd/1
66
Fractions within a fraction
Step 1-find the lcd of all 4 terms Step
2-multiply each term by the lcd/1
x
1
Lcd x2
1
x
67
Example
68
Solution lcd b2
69
Solution lcd b2
b
b
70
Next example
71
lcd ab
But this one needs to be reduced!
72
lcd ab
73
Exit ticket Simplify
74
Solving Rational equations
Do now page 60 11,12
Students will solve rational equations,
exploring two methods that develop the skills
learned for adding and multiplying rational
expressions, and monitoring for the creation of
extraneous solutions. This will be evidenced by
an exit problem.
75
Solving rational equations using the lcd method
How is this different than the ones you just
solved?
  1. Find the lcd of all terms
  2. Multiply each term by the lcd
  3. Solve the equation

STEPS
76
Solution
2
3
6 2a 9 6a
Look, we eliminated denominators!
77
6a 6a
-6
-6
____ ___ 8 8
78
Example
79
Lcd2x
4x 6 10 4x 4 x 1
80
Try this
81
Lcda(a2)
a23a64 a23a10 a2-3a-100 (a-5)(a2)0 a5,
a-2
82
Extraneous roots
Sometimes, when we check roots in the
original Equation, we arrive at an undefined
denominator. These are called extraneous
roots. Check the roots in the previous
problem Which one is extraneous? Why?
83
Extraneous roots
Sometimes, when we check roots in the
original Equation, we arrive at an undefined
denominator. These are called extraneous
roots. Check the roots in the previous
problem Which one is extraneous? Why? -2 is
extraneous because it made part of the equation
undefined, so the solution is 5.
84
Review
Today we will review simplifying rational
expressions and solving rational equations for
your test tomorrow.
Do Now Solve for x
85
Review Rationals-index card review problems
Multiply and express in simplest form
1.
For what value of x is this undefined?
2.
86
Review Rationals
Add or subtract and express in simplest form
3.
4.
5.
Express this complex fraction in simplest form
87
Solving
6.
88
Answers
Add corrected problems to index card for folder
89
Just Find the LCD
It is sometimes necessary to factor the
denominators!
90
Just Find the LCD
It is sometimes necessary to factor the
denominators!
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