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Chapter 6
Rational Expression and Equations
How are rational expressions simplified
and rational equations solved?
ACTIVATION
• Review Yesterday’s Warm-up
6-1 Multiplying and Simplifying
• EQ:
How do you multiply and
simplify rational expressions?
• Simplify the following
9
12
8
48
• Is the following a valid fraction?
• Why or Why not?
• Simplest form—when all common factors have been
removed
• Example
y = x2 + 10x + 25
x2 + 9x + 20
If the denominator cannot equal zero what do we do with
when we have a variable?
Multiplying Expressions
Simplify:
2x2 + 7x + 3
x-4
x2 - 16
x2 + 8x + 15
Dividing Expressions
• Invert the fraction behind the division
sign and multiply
6x + 6y
x–y
÷
18
5x – 5y
Homework
• PAGE(S): 248
• NUMBERS: 2 – 32 evens
ACTIVATION
• Review Yesterday’s Warm-up
6-2 Addition and Subtraction
• EQ:
How do you add and subtract
rational expressions?
Activation
What is required to add and subtract fractions?
Add
1 + 3
5
8
Add
1
x+1
+ 3
x -1
.
Examples
1
3

2
x 4 x2
Examples
Homework
• PAGE(S): 253 - 254
• NUMBERS: 2 – 30 evens
ACTIVATION
• Review Yesterday’s Warm-up
6-3 Complex Rational Expressions
• EQ:
How do you simplify complex
rational expressions?
How do you simplify the following:
1
2
4
5
Evaluating Complex Expressions
• Is nothing more than dividing fractions
x2
3
x2  4
6
x2 x 4

3
6
2
means
Homework
• PAGE(S): 258
• NUMBERS: 6 – 20 even
ACTIVATION
• Review Yesterday’s Warm-up
6-4 Division of Polynomials
• EQ:
How do you divide polynomials?
What procedures would you use to
solve the following problem:
34 4786
Can we translate this to algebraic
equations
x  4 x2  6x  8
• Example 2
x 2  4 x  4 x 4  6 x 2  8x  10
Homework
• PAGE(S): 262
• NUMBERS: 12 – 22 even
ACTIVATION
• Review Yesterday’s Warm-up
6-5 Synthetic Division
• EQ:
•
What is synthetic division?
Long division can be cumbersome
Patterns were seen that can be used
when the divisor is linear
(x3 +3x2 – x – 3) ÷ (x – 1)
Long division can be cumbersome
Patterns were seen that can be used
when the divisor is linear
(x3 +3x2 – x – 3) ÷ (x – 1)
The remainder theorem helps to
determine roots as well but does not give
the remaining factors/roots
Example:
Given: f(x)= x3 + 4x2 + 4x
are 2, -1 or 0 roots?
Homework
• PAGE(S): 265
• NUMBERS: 2 -12 even
ACTIVATION
• Review Yesterday’s Warm-up
6-6 Solving Rational Equations
• EQ:
How do you solve a rational
equation?
Solve
5 = 15 .
2x -2 x2 – 1
Solve
x = x+6.
x-1
x+3
Check for any values that cause
the fraction to be undefined
Check for any values that cause
the fraction to be undefined
• Example
x 1
4

x 5 x 5
• Remember to check for extraneous values
• Example
2
1
16

 2
x  5 x  5 x  25
• Remember to check for extraneous values
Homework
• PAGE(S): 269
• NUMBERS: 4, 8, 12, 14, 24, 26
ACTIVATION
• Review Yesterday’s Warm-up
6-7 Using Rational Equations
• EQ:
How do you translate word
problems into rational
equations that can be solved?
Examples:
Antonio, an experienced shipping clerk, can fill a
certain order in 5 hours. Brian a new clerk, needs
9 hours to do the same job. Working together,
how long would it take them to fill the order?
Work problems use inverses:
Antonio: 5 hrs
Brian: 9 hrs
Total job: t hrs
• Example
The speed of the stream is 4 km/hr. A
boat travels 6 km upstream in the same
time it takes to travel 12 km
downstream. What is the speed of the
boat in still water?
Distance
Rate
Time
Upstream
6 km
X–r
T
downstream
12
X+r
T
Homework
• PAGE(S): 273 - 275
• NUMBERS: 2, 4, 6, 14, 20
6-8 Formulas
• EQ:
How do you solve rational formulas for
a specified variable?
• Example
w1 d1
solve

for d1
w2 d 2
• Example
1
1
solve S  v1t  v 2 t for t
2
2
Homework
• PAGE(S): 278
• NUMBERS: 4, 8, 12, 16
6-9 Variation and Problem Solving
• EQ:
What are direct and inverse variation?
• Vocabulary
• Direct variation—when the ratio of
two numbers is constant
•
y = kxn
• Inverse variation—when the
product of a series of numbers is
constant
•
y=
k
xn
• Joint variation—multiple direct
variations
• k—the constant of variation
Y varies directly with the square of
x. What is the value of y when x =
3, if x=2 when y = 12.
y = kxn
• Example
Y varies inversely with the square of x. What is the
value of y when x = 3. If x=2 when y = 9
y= k
xn
Example:
• Example
Set up the following problem. Y varies inversely as x but directly as
the cube of v.
What is the value of y when x = 2 and v = 3, if y = 16 when x=3 and v = 2
Joint Variation
• when y varies jointly with x and the square
of z. Find the general equation if y= 12
when x = 2 and z = 3.
Homework
• PAGE(S): 283
• NUMBERS: 4 – 24 by 4’s
Homework
• PAGE(S): 289
• NUMBERS: all