Chapter 6 Rational Expressions and Equations

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Chapter 6Rational Expressions and Equations

• Section 6.1
• Multiplying Rational Expressions

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HW 6.1Pg 248 1-37Odd, 40-43
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Chapter 6Rational Expressions and Equations

• Section 6.2
• Addition and Subtraction

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LOGICAL REASONING Tell whether the statement is
always true, sometimes true, or never true.
Explain your reasoning.

• The LCD of two rational expressions is the
  product of the denominators.
• Sometimes
• The LCD of two rational expressions will have a
  degree greater than or equal to that of the
  denominator with the higher degree.
• Always

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Simplify the expression.
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HW 6.2 Pg 253-254 3-30 Every Third Problem
31-45 Odd
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Chapter 6Rational Expressions and Equations

• 6.3
• Complex Rational Expressions

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HW 6.3Pg 258 1-23 Odd, 26-28
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HW Quiz 6.3Monday, July 02, 2012
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Chapter 6Rational Expressions and Equations

• 6.4
• Division of Polynomials

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• Do a few examples of a poly divided by a monomial
• Discuss the proof of the remainder theorem

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HW 6.4Pg 262 1-25 Odd, 26-32
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Chapter 6Rational Expressions and Equations

• Section 6.5
• Synthetic Division

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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

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Part 1
Dividing using Synthetic
Division
EXAMPLE 1
To see how synthetic division works, we will use
long division to divide the polynomial
by
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Dividing PolynomialsUsing Synthetic Division
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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).
1
- 3
1 6 8 -2
- 3
Add these up
- 9
3
Add these up
Add these up
1
3
- 1
1
x2 x
This is the remainder
Put variables back in (one x was divided out in
process so first number is one less power than
original problem).
List all coefficients (numbers in front of x's)
and the constant along the top. If a term is
missing, put in a 0.
So the answer is
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Let's try another Synthetic Division
0 x3
0 x
1
4
1 0 - 4 0 6
4
Add these up
16
48
192
Add these up
Add these up
Add these up
1
4
12
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This is the remainder
x3 x2 x
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Now put variables back in (remember one x was
divided out in process so first number is one
less power than original problem so x3).
List all coefficients (numbers in front of x's)
and the constant along the top. Don't forget the
0's for missing terms.
So the answer is
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Let's try a problem where we factor the
polynomial completely given one of its factors.
You want to divide the factor into the polynomial
so set divisor 0 and solve for first number.
- 2
4 8 -25 -50
- 8
Add these up
0
50
Add these up
Add these up
No remainder so x 2 IS a factor because it
divided in evenly
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0
- 25
0
x2 x
Put variables back in (one x was divided out in
process so first number is one less power than
original problem).
List all coefficients (numbers in front of x's)
and the constant along the top. If a term is
missing, put in a 0.
So the answer is the divisor times the quotient
You could check this by multiplying them out and
getting original polynomial
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HW 6.5Pg 265 1-19
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6-6 Solving Rational Equation
. . . And Why To solve problems using rational
equations
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To solve a rational equation, we multiply both
sides by the LCD to clear fractions.
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Multiplying by the LCD
Multiplying to remove parentheses
Simplifying
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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.
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No Solution
y 57
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The LCD is x - 2. We multiply by x - 2.
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The number -2 is a solution, but 2 is not since
it results in division by O.
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The solutions are 2 and 3.
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h. x 1, -½
f. x -3, 4
g. x 1, -½
e. x 3
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This checks in the original equation, so the
solution is 7.
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x 7
x -13
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HW 6.6Pg 269 1-25 Odd, 26-34
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6-7
Warm Up
Solve the following equation
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Tom knows that he can mow a golf course in 4
hours. He also knows that Perry takes 5 hours to
mow the same course. Tom must complete the job in
2! hours. Can he and Perry get the job done in
time? How long will it take them to complete the
job together?
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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.

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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 Question How long will it
take the two of them to mow the lawn together?
Data Tom takes 4 hours to mow the lawn. Perry
takes 5 hours to mow the lawn.
Tom can do 1/4 of the job in one hour
Perry can do 1/5 of the job in one hour
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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 Let t represent the
total number of hours it takes them working
together. Then they can mow 1/t of it in 1 hour.
Translate to an equation.
Tom can do 1/4 of the job in one hour
Together they can do 1/t of the job in one hour
Perry can do 1/5 of the job in one hour
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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?
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Tom knows that he can mow a golf course in 4
hours. He also knows that Perry takes 5 hours to
mow the same course. Tom must complete the job in
2! hours. Can he and Perry get the job done in
time? How long will it take them to complete the
job together?
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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.
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Objective Solve work problems using rational
equations.
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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.
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Objective Solve motion problems using rational
equations.
r 36 km/h
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Objective Solve motion problems using rational
equations.

• Try This
• 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.
• 35.5 mi/h
• 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
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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.

• 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 Jorges trip will take.

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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.

• Write a rational expression, in terms of d and r,
  that gives his average speed for the entire trip.

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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.

• 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.

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HW 6.7 Pg 273 1-27 Odd, 29-33
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6-8
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We solve the formula for the unknown resistance
r2.
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We solve the formula for the unknown resistance
r2.
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HW 6.8Pg 278 1-30
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6-9

• What you will learn
• Find the constant and an equation of variation
  for direct and joint variation problems.
• To find the constant and an equation of variation
  for inverse variation problems
• To solve direct, joint, and inverse variation
  problems

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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
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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.
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Objective Find the constant of variation and an
equation of variation for direct variation
problems.
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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.
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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)
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Objective Find the constant of variation and an
equation of variation for joint variation
problems.
Try This
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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.
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Objective Find the constant of variation and an
equation of variation for inverse variation
problems.
EXAMPLE 3
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Objective Find the constant of variation and an
equation of variation for inverse variation
problems.
EXAMPLE 3
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Objective Find the constant of variation and an
equation of variation for inverse variation
problems.
Try This
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Describe the variational relationship between x
and z and demonstrate this relationship
algebraically.

• x varies directly with y, and y varies inversely
  with z.
• x varies inversely with y, and y varies inversely
  with z.
• x varies jointly with y and w, and y varies
  directly with z, while w varies inversely with z.

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The weight of an object on a planet varies
directly with the planets 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

• 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.
• Earth has a radius of 6378 km, while Mercury
  (whose density is almost the same as Earths) has
  a radius of 4878 km. If you weigh 125 lb on
  Earth, how much would you weigh on Mercury?

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HW 6.9 Pg 283-284 1-32
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Chapter 6

• Review

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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

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Simplify
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Simplify
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Simplify
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Simplify
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Simplify
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Simplify
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Simplify
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Simplify
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Solve
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Solve
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Divide
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Divide
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HW R-6 Pg 287-288 1-29