Transcript Chapter 6 Rational Expressions and Equations

Chapter 6
Rational Expressions and Equations
Section 6.1
Multiplying Rational Expressions
HW #6.1
Pg 248 1-37Odd, 40-43
Chapter 6
Rational Expressions and Equations
Section 6.2
Addition and Subtraction
9.
10.
11.
12.
13.
14.
15.
16.
9.
13.
11.
10.
14.
15.
12.
16.
LOGICAL REASONING Tell whether the statement is
always true, sometimes true, or never true. Explain your
reasoning.
1. The LCD of two rational expressions is the product of the
denominators.
• Sometimes
2. The LCD of two rational expressions will have a degree
greater than or equal to that of the denominator with the
higher degree.
• Always
Simplify the expression.
17.
18.
19.
20.
HW #6.2
Pg 253-254 3-30 Every Third
Problem 31-45 Odd
Chapter 6
Rational Expressions and Equations
6.3
Complex Rational Expressions
HW 6.3
Pg 258 1-23 Odd, 26-28
HW Quiz 6.3
Monday, March 28, 2016
f ( x  h)  f ( x )
1
Evaluate
for f ( x) 
h
x
Chapter 6
Rational Expressions and Equations
6.4
Division of Polynomials
• Do a few examples of a poly divided by a
monomial
• Discuss the proof of the remainder theorem
HW #6.4
Pg 262 1-25 Odd, 26-32
Chapter 6
Rational Expressions and Equations
Section 6.5
Synthetic Division
Part
1
Dividing using Synthetic Division
Objective: Use synthetic division to find the quotient of certain polynomials
• Algorithm
– A systematic procedure for doing certain computations.
• The Division Algorithm used in section 6.4 can be
shortened if the divisor is a linear polynomial
– Synthetic Division
Part
1
Dividing using Synthetic Division
EXAMPLE 1
To see how synthetic division works, we will use long division to divide
the polynomial 2 x 3  x 2  3 by x  3
Dividing
Polynomials
Using
Synthetic
Division
Synthetic Division
There is a shortcut for long division as long as the divisor is x – k
where k is some number. (Can't have any powers on x).
Set divisor = 0 and
3
2
1 x  6 x  8x  2
solve. Put answer
here.
x3
x + 3 = 0 so x = - 3
-3
1
6
8
-2
up3these
Bring
number
down
below
Addupthese
line up
- 3 firstAdd
- 9theseAdd
Multiply
Multiply
these
these and
and
2 +3 x - 1
This is the remainder
1
x
1
put
answer
put answer
above
line
above line
Put
variables
back
in (one
was of
divided
outthe
in
Sonext
the
Listanswer
all
coefficients
is:
(numbers
in xfront
x's) and
in
in next process so first number is one less power than
2 top. If a term is missing, put in a 0.
constant along the
column
column original problem).
1
x  3x  1 
x3
Let's try another Synthetic Division
Set divisor = 0 and
solve. Put answer
here.
4
1
0 x3
0x
1 x  4x  6
4
2
x4
x - 4 = 0 so x = 4
0
-4
0
6
up48
Bring
number
down
these
below
Add
upthese
line
Add
up these up
4 firstAdd
16theseAdd
192
Multiply
Multiply
Multiply
these
these and
and
3 + 4 x2 + 12 x + 48 198
This is the
these
and
1
x
put
answer
put answer
remainder
put
answer
above
line
above line Now put variables back in (remember one x was
above
lineanswer
Sonext
the
List
all coefficients
is:
(numbers in front of x's) and the
in
in next divided out 3in process2so first number is one less
in next
constant along the top. Don't forget the 0's for missing
column
column power than original problem so x3).
column
terms.
198
x  4 x  12 x  48 
x4
Let's try a problem where we factor the polynomial
completely given one of its factors.
4 x 3  8 x 2  25 x  50
-2
4
factor : x  2
You want to divide
the factor into the
polynomial so set
divisor = 0 and solve
for first number.
8 -25 -50
up
Bring
number
down
below
Addupthese
line up
- 8 firstAdd
0theseAdd
50these
Multiply
Multiply
these
No remainder so x + 2
these and
and
2
4 x + 0 x - 25
0
put
IS a factor because it
put answer
answer
above
line
divided in evenly
above line
Put
variables
back
in
(one
x
was
divided
outthe
in
Sonext
the
Listanswer
all coefficients
is the divisor
(numbers
times in
thefront
quotient:
of x's) and
in
in next
process
sothe
first
number
is one
less power
You could
check
this
byIf a term
constant
along
top.
is missing,
putthan
in a 0.
column
2
column
original
problem).
multiplying
them out
and getting
original polynomial
x  24 x
 25

HW #6.5
Pg 265 1-19
. . . And Why
To solve problems using rational equations
A rational equation is an equation that contains one or more rational
expressions. These are rational equations.
To solve a rational equation, we multiply both sides by the
LCD to clear fractions.
Multiplying by the LCD
Multiplying to remove parentheses
Simplifying
2
x=
3
120
x=11
The LCD is x - 5, We multiply by x - 5 to clear fractions
5 is not a solution of the original equation
because it results in division by 0, Since 5 is
the only possible solution, the equation has
no solution.
y = 57
No Solution
The LCD is x - 2. We multiply by x - 2.
The number -2 is a solution, but 2 is not since it results in
division by O.
The solutions are 2 and 3.
e. x = 3
f. x = -3, 4
g. x = 1, -½
h. x = 1, -½
This checks in the original equation, so the
solution is 7.
x=7
x = -13
HW #6.6
Pg 269 1-25 Odd, 26-34
Warm Up
Solve the following equation
knows
he canmower
mow
IfTom
Perry
getsthat
a larger
a golf
in mow
4 hours.
so
thatcourse
he can
theHe
also knows
thatinPerry
takes 5
course
alone
3 hours,
hourslong
to mow
same
how
will itthetake
Tom
course.
Tom
complete
and
Perry
tomust
complete
the
the together?
job in 2! hours. Can he
job
and Perry get the job done in
time? How long will it take
them to complete the job
together?
Solving Work Problems
If a job can be done in t hours, then 1/t of it
can be done in one hour. This is also true for
any measure of time.
Objective: Solve work problems using rational equations.
Tom can mow a lawn in 4 hours. Perry can mow the
same lawn in 5 hours. How long would it take both of
them, working together with two lawn mowers, to
mow the lawn?
UNDERSTAND the problem
Data: Tom takes 4 hours to
mow the lawn. Perry takes 5
Question: How long will it take
hours to mow the lawn.
the two of them to mow the lawn
together?
Tom can do 1/4 of the job in one hour
Perry can do 1/5 of the job in one hour
Objective: Solve work problems using rational equations.
Tom can mow a lawn in 4 hours. Perry can mow the
same lawn in 5 hours. How long would it take both of
them, working together with two lawn mowers, to
mow the lawn?
Develop and carryout a PLAN
Translate to an equation.
Let t represent the total number of
hours it takes them working
together. Then they can mow 1/t
of it in 1 hour.
Tom can do 1/4 of the job in
oneTogether
hour they can do 1/t of
the job in one hour
Perry can do 1/5 of the job
in one hour
1 1 1
 
4 5 t
Objective: Solve work problems using rational equations.
Tom can mow a lawn in 4 hours. Perry can mow the
same lawn in 5 hours. How long would it take both of
them, working together with two lawn mowers, to
mow the lawn?
knows
he canmower
mow
IfTom
Perry
getsthat
a larger
a golf
in mow
4 hours.
so
thatcourse
he can
theHe
also knows
thatinPerry
takes 5
course
alone
3 hours,
hourslong
to mow
same
how
will itthetake
Tom
course.
Tom
complete
and
Perry
tomust
complete
the
the together?
job in 2! hours. Can he
job
and Perry get the job done in
1 long
1 will it take
time?1 How
 
3 t
them4to complete
the job
together?
5
t = 1 hours
12
2
t  2 hours
5
Objective: Solve work problems using rational equations.
At a factory, smokestack A pollutes the air twice as
fast as smokestack B.When the stacks operate
together, they yield a certain amount of pollution in
15 hours. Find the time it would take each to yield
that same amount of pollution operating alone.
1/x is the fraction of the pollution produced by A in 1 hour.
1/2x is the fraction of the pollution produced by B in 1 hour.
1/15 is the fraction of the total pollution produced by A and B in 1 hour.
1 1
1
+
=
x 2x 15
Objective: Solve work problems using rational equations.
A  32 hours,B  96 hours
An airplane flies 1062 km with the wind. In the
same amount of time it can fly 738 km against
the wind. The speed of the plane in still air is
200 km/h. Find the speed of the wind.
Objective: Solve motion problems using rational equations.
r = 36 km/h
Objective: Solve motion problems using rational equations.
Try This
d. A boat travels 246 mi downstream in the same time it takes to
travel 180 mi upstream. The speed of the current in the stream is
5.5 mi/h. Find the speed of the boat in still water.
a. 35.5 mi/h
e. Susan Chen plans to run a 12.2 mile course in 2 hours. For the
first 8.4 miles she plans to run at a slower pace, then she plans to
speed up by 2 mi/h for the rest of the course. What is the slower
pace that Susan will need to maintain in order to achieve this
goal?
e. about 5.5 mi/h
Try This
Jorge Martinez is making a business trip by car. After driving
half the total distance, he finds he has averaged only 20 mi/h,
because of numerous traffic tie-ups. What must be his average
speed for the second half of the trip if he is to average 40 mi/h
for the entire trip? Answer this question using the following
method.
1. Let d represent the distance Jorge has traveled so far,
and let r represent his average speed for the
remainder of the trip. Write a rational function, in terms
of d and r, that gives the total time Jorge’s trip will
take.
Try This
Jorge Martinez is making a business trip by car. After driving
half the total distance, he finds he has averaged only 20 mi/h,
because of numerous traffic tie-ups. What must be his average
speed for the second half of the trip if he is to average 40 mi/h
for the entire trip? Answer this question using the following
method.
2. Write a rational expression, in terms of d and r, that
gives his average speed for the entire trip.
Try This
Jorge Martinez is making a business trip by car. After driving
half the total distance, he finds he has averaged only 20 mi/h,
because of numerous traffic tie-ups. What must be his average
speed for the second half of the trip if he is to average 40 mi/h
for the entire trip? Answer this question using the following
method.
3. Using the expression you wrote in part (b), write an
equation expressing the fact that his average speed
for the entire trip is 40 mi/h. Solve this equation for r if
you can. If you cannot, explain why not.
HW #6.7
Pg 273 1-27 Odd, 29-33
PV
T 
K
We solve the formula for the unknown resistance r2.
We solve the formula for the unknown resistance r2.
HW #6.8
Pg 278 1-30
What you will learn
1. Find the constant and an equation of variation for direct and
joint variation problems.
2. To find the constant and an equation of variation for inverse
variation problems
3. To solve direct, joint, and inverse variation problems
Objective: Find the constant of variation and an equation of variation for
direct variation problems.
Direct Variation
Whenever a situation translates to a linear
function f(x) = kx, or y = kx, where k is a
nonzero constant, we say that there is direct
variation, or that y varies directly with x. The
number k is the Constant of Variation
Objective: Find the constant of variation and an equation of variation for
direct variation problems.
The constant of variation is 16.
The equation of variation is y = 16x.
Objective: Find the constant of variation and an equation of variation for
direct variation problems.
Objective: Find the constant of variation and an equation of variation for joint
variation problems.
Joint Variation
y varies jointly as x and z if there is some
number k such that y = kxz, where
k  0, x  0, and z  0.
Objective: Find the constant of variation and an equation of variation for joint
variation problems.
EXAMPLE 2
Suppose y varies jointly as x and z. Find the constant of variation
and y when x = 8 and z = 3, if y = 16 when z = 2 and x = 5.
Find k
y = kxz
16 = k(2)(5)
16 8
k 
10 5
8
y  xz
5
8
y   8  3
5
y
192
5
Objective: Find the constant of variation and an equation of variation for joint
variation problems.
Try This
Objective: Find the constant of variation and an equation of variation for inverse
variation problems.
Inverse Variation
y varies inversely as x if there is some
number k such that y = k/x, where
k  0 and x  0.
Objective: Find the constant of variation and an equation of variation for inverse
variation problems.
EXAMPLE 3
Objective: Find the constant of variation and an equation of variation for inverse
variation problems.
EXAMPLE 3
Objective: Find the constant of variation and an equation of variation for inverse
variation problems.
Try This
1
30
Describe the variational relationship
between x and z and demonstrate this
relationship algebraically.
1. x varies directly with y, and y varies inversely with z.
2. x varies inversely with y, and y varies inversely with z.
3. x varies jointly with y and w, and y varies directly with
z, while w varies inversely with z.
The weight of an object on a planet varies directly with the
planet’s mass and inversely with the square of the planet's
radius. If all planets had the same density, the mass of the planet
would vary directly with its volume, which equals
4 3
r
3
1. Use this information to find how the weight of an
object w varies with the radius of the planet,
assuming that all planets have the same density.
2. Earth has a radius of 6378 km, while Mercury
(whose density is almost the same as Earth’s)
has a radius of 4878 km. If you weigh 125 lb on
Earth, how much would you weigh on Mercury?
HW #6.9
Pg 283-284 1-32
Chapter 6
Review
Two Parts
Part 1
•
•
•
•
•
Add/Subtract/Multiply/Divide Rational Expressions
Solve Rational Equations
Long Division/Synthetic Division
Direct/Joint/Inverse Variation
Challenge Problems
Part 2
•
•
•
•
Work Problems
Distance Problems
Problems with no numbers
Challenge Problems
Simplify
32 x  64 x  24 x
4
4x
6
5
4
Simplify
Simplify
Simplify
Simplify
Simplify
Simplify
Simplify
x 1
x 1
7
Solve
Solve
Divide
Divide
Compute the value of
f  x  h  f  x 
h
for
1
f  x 
x2
Find the value of k if (x + 2) is a factor of
x  kx  5 x  6
3
2
HW # R-6
Pg 287-288 1-29