Please
CLOSE
YOUR LAPTOPS,
and turn off and put away your
cell phones,
and get out your notetaking materials.
Coming up:
• Today: (Monday, 11/25)
• Hand back/review Test 3
• Lecture on Section 7.1/2 (Rational expressions)
• Daily Quiz 33 at end of hour on material in today’s
lecture
• Tomorrow: (Tuesday, 11/26)
•
•
•
•
HW 38 due on section 7.1/2
Practice Quiz 9 due
Lecture on Section 10.1 (Radical functions)
Take Weekly Quiz 9 on Sec. 7.1 and Test 3
• Monday, 12/2:
• HW 39 due on section 10.1
• Daily Quiz on section 10.1
• Lecture on Section 10.2
Test 3 Results:
•Average class score after partial credit: __________
•Commonly missed questions: #_________________
Grade Scale
Grade
A
A-
B+
B
B-
C+
C
C-
F
Points
≥ 920 ≥ 890 ≥ 860
≥ 830
≥ 800
≥ 750
≥ 700 ≥ 670 < 670
% Score
≥ 92
≥ 83
≥ 80
≥ 75
≥ 70
≥ 89
≥ 86
≥ 67
< 67
If you got less than 75% on Test 1, make sure to go over your test
with me or a TA sometime in the next few days. This material will
be used in the next unit, and it will also be covered again on the
final exam.
Teachers: You can insert screen shots of any test problems you want to go over with your students here.
Sections 7.1/2
Rational Expressions
Remember this problem from Test 3?
In the last chapter, you learned how to do this
problem using long division. Today we’ll look at
an alternate way to do problems like this (with no
remainder) using factoring.
Rational Expressions
Rational expressions are ratios of two polynomials,
just like a rational number is a ratio of two integers.
Examples of Rational Expressions:
3x 2  2 x  4
4x  5
4x  3y
2
2
2 x  3 xy  4 y
3x 2
4
For this last one, remember
that 4 can be considered a
polynomial of degree 0.
Rational expressions can be simplified, multiplied, divided,
added and subtracted using factoring methods similar to the
ones we use with regular fractions (rational numbers).
• Simplifying a rational expression means writing it in lowest terms
or simplest form.
• To do this, we need to use the Fundamental Principle of
Rational Expressions:
If P, Q, and R are polynomials, and Q and R are not 0, then PR  P
QR Q
• This is similar to what you do when you simplify
a rational number (fraction):
Example: Simplify 105
49
Solution: First factor the numbers:
105 = 5*21 = 5*3*7
49 = 7*7
Next, rewrite the ratio in its factored form:
105 = 5*3*7
49
7*7
Finally, cancel the common factors and rewrite in simplified form:
105 = 5*3*7 = 5*3 = 15
49
7*7
7
7
Simplifying a Rational Expression:
1) Completely factor the numerator and denominator
polynomials.
2) Apply the Fundamental Principle of Rational
Expressions to cancel common factors in the
numerator and denominator.
Warning! DO NOT multiply out the factors at the end
like you did with the numbers in a simplified fraction.
Warning 2! Only common FACTORS can be canceled
from the numerator and denominator. Make sure any
expression you eliminate is a factor, not just a term
within a factor.
Example
Simplify the following expression.
7( x  5) 7
7 x  35


2
x( x  5)
x  5x
x
Question:
Is the following simplification correct?
2 x  7 2 x /7 2/x 2



5 x  7 5 x /7 5/x 5
Answer:
NO!!!!
Remember, we can only cancel entire
FACTORS, not terms with factors.
2x+7 is a factor; it could be written as (2x+7).
The 2x and the 7 are terms in the factor.
The following example with numbers illustrates the
error in the previous case of “bad cancelling”:
Correct:
27 9 3
 
5  7 12 4
Would we get the same answer if we
cancelled the 7’s first?
Incorrect!!
/
/
3

4
Problem from today’s homework:
Solve by factoring
both trinomials and
then canceling any
common factors.
+ 3)(3x /– 1)
(x – 2)(3x /– 1)
(x
x+3
x-2
Example
Simplify the following expression.
7  y  1( y  7)


y 7
y 7
1
Revisiting this problem from Test 3:
Instead of using long division, let’s try
factoring and canceling:
HINT: Use a3 – b3 = (a – b)(a2 + ab + b2)
This looks like ( )3 - ( )3. What goes in the
parentheses?
a = x and b = 2.
x3 – 8 = (x – 2)(x2 + 2x + 4).
So x3 – 8 = (x –/ 2)(x2 + 2x + 4) = x2 + 2x + 4
(x – 2)
(x /– 2)
Note that this is the same answer we
would have gotten using long division.
To evaluate a rational expression for a
particular value of a variable, substitute the
replacement value into the rational expression
in place of that variable and simplify the result.
Example
Evaluate the following expression for y = -2.
22
y2
4 4



 5  y  5  (2)  7 7
In the previous example, what would happen
if we tried to evaluate the rational
expression for y = 5?
52
3
y2


5 y 5 5 0
This expression is undefined!
• We have to be able to determine when a
•
•
rational expression is undefined.
A rational expression is undefined when the
denominator is equal to zero.
The numerator being equal to zero is okay
(the rational expression simply equals zero).
Example Find any real numbers that make the
following rational expression undefined.
9x  4x  7
2
15 x  45 x
3
The expression is undefined when 15x2 + 45 = 0.
Factoring this gives 15x(x + 3) = 0,
so the expression is undefined when x = -3 or x = 0.
The set of numbers for which an expression is
defined is called the domain of the expression.
The domain is written in set notation.
The domain for the expression in this example would be:
{ x | x ≠ 0, -3}
Example 2:
Find the domain of the expression
x 2  3x  4
x 2  x  20
Solution: Set the denominator equal to zero and
solve for x.
x2 – x – 20 = 0
(x – 5)(x + 4) = 0
(x – 5) = 0 gives x = 5,
and (x + 4) = 0 gives x = -4
Therefore the domain is {x | x ≠ 5 ,-4}
Problem from today’s homework:
0, 2,-1
To answer this question, you need to find all solutions of the equation
obtained by setting the denominator equal to zero.
•
•
•
•
(Notice that in DOMAIN questions, you focus only on the DENOMINATOR).
How many solutions are you expecting to find? Why?
Answer: At most three, because the polynomial is cubic, i.e. has a degree of three.
Find the answers by factoring the polynomial and setting each of the three factors to
equal to zero.
• Factoring: x(x - 2)(x + 1)
• Solutions: x = 0, x = 2, x = -1. These are the numbers that are NOT in the domain.
Check: Plug each of these three numbers back into the denominator
of the function and show that each one gives zero as the result.
Steps in multiplying two or more
rational expressions:
1. Factor all numerators and denominators.
2. Cancel any common factors between all
numerators and all denominators.
3. Don’t multiply out any polynomial terms in
the answer – just leave them factored.
IMPORTANT: Always factor and cancel first
instead of multiplying first, even though
the directions say “multiply”.
Example
( x  2) 2
5

Multiply: 
10
2x  4
( x  2)( x  2)  5

5  2  2  ( x  2)

( x  2) ( x  2)  5
5  2  2  ( x  2)
x2

4
When dividing rational expressions, first
change the division into a multiplication
problem, where you use the reciprocal of
the divisor as the second part of the product.
Then treat it as a multiplication problem
(factor, multiply, simplify).
Example
Divide the following rational expression.
25
( x  3) 5 x  15 ( x  3)




5
5 x  15
5
25
2
( x  3)(x  3)  5  5
 x3
5  5( x  3)
2
REMINDER:
The assignment on today’s material (HW 38) is due at
the start of the next class session.
Please open your laptops and start working on the
homework assignment if there’s any time left before it’s
time to take the quiz at the end of the class session.
You can keep your yellow formula sheets out when you take
the quiz.
Lab hours in 203:
Mondays through Thursdays
8:00 a.m. to 7:30 p.m.
Please remember to sign in on the Math 110 clipboard
by the front door of the lab
Please open Daily Quiz 33.
If you have any time left after
finishing the quiz problems,
CHECK YOUR FACTORING ANSWERS
before you submit the quiz.
• A scientific calculator may be used on this quiz.
• Remember to turn in your answer sheet to the TA
when the quiz time is up.