Math 002 – Student Notes and Assignments
Unit 3 – Factoring Polynomials & Rational Expressions and Equations
Unit 3
5.5
5.6
BOOK
BOOK
MML
BOOK
BOOK
MML
Homework
pg. 366: 6, 38, 42, 50, 56, 60
pg. 376: 4, 6, 12, 20, 26, 76
#11 (5.5, 5.6)
pg. 382: 4, 8, 22, 24, 54, 80
pg. 396: 30, 36, 48, 78, 88
#12 (5.7, 5.8)
6.1
BOOK
pg. 420: 6, 14, 30, 36, 48, 83
6.2
BOOK
MML
MML
BOOK
BOOK
MML
BOOK
BOOK
MML
pg. 429: 18, 36, 50, 52,103
Q3 (5.5-5.8)
#13 (6.1, 6.2)
pg. 445: 16, 26, 30, 34, 98
pg. 452: 6, 12, 18, 22, 28, 46
#14 (6.4, 6.5)
pg. 462: 28, 30, 34, 38, 40
pg. 470: 6, 12, 18, 22, 38, 40
#15 (6.6)
5.7
5.8
6.4
6.5
6.6
6.7
EXAM 3
11-Mar
Topic
Factoring by GCF & Grouping
Factoring Trinomials
Due Date
M: Mar 25
Factoring by Special Products
Solve by Factoring
TH: Mar 28
Rational Expressions: Simplify,
Multiply & Divide
Rational Expressions: Add/Sub
M: Apr 1
M: Apr 1
Dividing Polynomials
Solving Equations
*FRI: Apr 5
Rational Eqns & Problem Solving
Variation and Problem Solving
M: Oct 29
TH-April 11 & FR – April 12
12-Mar
13-Mar
14-Mar
15-Mar
Week 8
§ 5.5, 5.6
§ 5.5, 5.6
18-Mar
19-Mar
20-Mar
§ 5.6
21-Mar
22-Mar
28-Mar
29-Mar
SPRING BREAK
25-Mar
26-Mar
27-Mar
MML #11 (5.5, 5.6)
MML #12 (5.7, 5.8)
Quiz 5 (5.5-5.8)
Quiz 5 (5.5-5.8)
RETAKE for Exam 1 or Exam 2
§ 6.1, 6.2
§ 6.1
Week 9
§ 5.7, 5.8
§ 5.7
1-Apr
§ 5.8
2-Apr
3-Apr
4-Apr
MML #13 (6.1, 6.2)
Week 10
MML Q3
§ 6.4, 6.5
§ 6.2
5-Apr
MML #14 (6.4, 6.5)
8-Apr
§ 6.4
9-Apr
Quiz 6 (6.2-6.4)
Quiz 6 (6.2-6.4)
§ 6.5, 6.6
§ 6.5
10-Apr
11-Apr
12-Apr
MML #15 (6.6)
Week 11
Exam 3
§ 6.6
Review
Review
§ 7.1, 7.2
Exam 3
§ 7.1, 7.2
1
Section 5.5 – The Greatest Common Factor and Factoring by Grouping
Objectives:
 Identify the GCF.
 Factor out the GCF of a polynomial’s terms.
 Factor by grouping.
Warm-up: Multiply:
Factoring is the reverse process of multiplying. It is the process of writing a polynomial as a product.
1. Finding the GCF, Greatest Common Factor, from a list of monomials. EX 1
Step 1:
Find the GCF of the numerical coefficients.
Step 2:
Find the GCF of the variable factors.
Step 3:
The product of the factors found in Steps1 and 2 is the GCF of the monomials.
Ex 1:
Find the GCF
2 3
a) 3x y , 9x4 y3
2
3
b) 6a , 12a b,
36ab
2. Factoring out the GCF of a polynomial’s terms. Use the distributive property in reverse. EX 2 – EX 5
a)
common monomial factor: __________
factor ______ out of each term
Use the distributive property in reverse.
b)
common monomial factor: __________
factor _______out of each term
Use the distributive property in reverse.
3. Factor out -1 from a polynomial.
a)  x  2 y
b)  3x  8z
c)  3r  2s  3
2
4. Factoring Polynomials by Grouping. Final answer must be a product. EX 7 – EX 10
a)
15xy  20 x  6 y  8
b)
20 xy  8x  5 y  2
c)
16ab  8a  6b  3
d)
xy  4 y  3x  12
FACTORING BY GROUPING
Sometimes it is possible to factor a polynomial by grouping terms of the polynomial and looking
for common terms in each group.
 Factor out the GCF of the polynomial.
 Group the polynomial into two binomials.
 Factor out the GCF from each binomial.
 Factor out the GCF of the entire polynomial.
3
Section 5.6 – Factoring Trinomials
Review. Recall that to multiply x  2 and x  3 , we proceed as follows:
1. Factoring trinomials of the form
: Leading coefficient 1. EX 1, EX 2, EX 3
x 2  7x  6
Consider the trinomial
Is there a common factor for all of the terms?
Leading coefficient 1?

Find all possible factor pairs of the constant .

Find the pair of factors whose sum is __________________.
Example 1
a)
Factor each trinomial if possible.
m2  3m  10
b)
c)
factors of ____:
factors of _____:
sum of factors = ____
sum of factors = ____
3y2  21y  18
d)
FACTORING A TRINOMIAL OF THE FORM
x 2  bx  c
If the trinomial has a leading coefficient of 1, follow these steps to factor:
 find all possible factor pairs of the constant c
 find the pair of factors whose sum is the middle term b
4
2. Factoring trinomials of the form
Use the
: Leading coefficient is not 1. EX 9, EX 10
method:
Step 1:
Find two numbers whose product is
Step 2:
Write the term
Step 3:
Factor by grouping.
Example 2
bx
ac
and whose sum is
b.
as a sum by using the factors found in Step 1.
Factor each trinomial if possible.
a)
b)
c)
d)
5
Section 5.7 – Factoring by Special Products
Some trinomials have special forms. If you can recognize them, factoring will be easier.
1. Square Trinomials:
When you square binomials, the resulting polynomials are called
perfect square trinomials.
2. Factoring a perfect square trinomial. EX 1, EX 2
Example 1: Factor completely.
a)
x2  30x  225
b)
x2  22x  121
c)
Generalize:
6
3. Factoring the difference of two squares. EX 3, EX 4
Recall: The product of conjugate pairs is the difference of squares.
Example 2:
Factor completely.
d)
e)
f)
g)
4. Factoring the sum of two squares.
5. Factoring the sum and difference of two cubes. EX 6, EX 7, EX 8, EX 9
 SOAP
x3  y 3 
x3  y 3 
Example 3:
h)
i)
j)
7
Steps for factoring a polynomial
1. Factor out all common factors.
2. If a polynomial has two terms, check for the following types:
a. The difference of two squares:
b. The sum of two squares:
c. The sum of two cubes:
d. The difference of two cubes:
3. If a polynomial has three terms, check for the following problem types:
a. A perfect square trinomial:
b.
4.
5.
6.
If the trinomial is not a perfect square, attempt to factor it as a general trinomial using
factoring by grouping.
If a polynomial has four or more terms, try factoring by grouping.
Continue until each individual factor is prime.
Check the results by multiplying.
Factoring Flowchart:
8
Section 5.8 – Solving Equations by Factoring and Problem Solving
1. Vocabulary:

A polynomial equation is the result of setting two polynomials equal _________________.

A polynomial equation is in standard form if one side of the equation is _______.
2. Zero-Factor Property. EX 1 – EX 7
Zero-Factor Property
If A and B are real numbers and
, then
or
This property is true for three or more factors also.

In other words, if the product of two or more real numbers is zero, then at least one number must
be zero.

This is the main idea of this section and a key concept in algebra.
Example 1: Solve.
What does this mean graphically?
9
3. Solving a Polynomial Equation by factoring: EX 1 – EX 7
Example 2: Solve.
a.
b.
Generalize
Step 1:
Write the equation in standard form ___________________________________________.
Step 2:
Factor the polynomial completely.
Step 3:
Set each factor containing a variable equal to 0.
Step 4:
Solve the resulting equations.
Step 5:
Check each solution in the original equation.
4. Finding -intercepts of polynomial functions.
(a) Match each function with its graph. EX 10

Solutions to a polynomials equation are also called ________________________.

Graphically, solutions to a polynomial function are where the graph
_____________________ _________________ or rather, _________________________.
(b) Write the factored form of an equation given solutions.
Example 3: Write the factored form of an equation which has the solutions -4, -1, and 8.
10
5. Solving problems modeled by polynomial equations.
(a) Using the Pythagorean Theorem. EX 9
(#78) The longer leg of a right triangle is 4 feet longer that the other leg. Find the length of
the two legs if the hypotenuse is 20 feet.
(b) Finding the return time of a rocket. EX 8
After t seconds, the height
given by the function
of a model rocket launched from the ground into the air is
.
1) How high is the rocket at
second?
2) Find how long it takes the rocket to reach a height of 96 feet.
3) When will the rocket hit the ground?
11
Sections 6.1 Rational Functions and Multiplying and Dividing Rational Expressions
Objectives:
 Simplify rational expressions.
 Multiply rational expressions.
 Divide rational expressions.
 Find the domain of a rational function
1. Define rational number:
Define rational expression:
Define rational function:
2. The domain of a rational function is defined for all real numbers except when
____________________________. EX 1, EX 2
Example 1: Find the domain of each rational expression.
a.
b.
c.
3. Fundamental Principle of Rational Expressions.
For any rational expression
and any polynomial R, where
,
 This principle states that when we multiply or divide the numerator AND denominator by the
same non-zero polynomial, we obtain an ___________________rational expression.
 Simplifying rational expressions is similar to ____________________________.
 This property allows us to divide out common factors, not terms.
Here’s an illustration of the difference.
12
4. Simplifying Rational Expressions. EX 3, EX 4, EX 5
Step 1:
Factor the numerator and denominator.
Step 2:
Divide out any factors that are common to both the numerator and denominator.
Example 2: Simplify each rational expression. Tip: Write factors in descending order,
alphabetical order, and leading coefficient positive.
a)
#15
b)

The following are equivalent:

The following are opposites:
_________
Tip: Write factors in descending order, alphabetical order, and leading coefficient positive.
#24

Signs of fractions/rational expressions:
13
5.
Multiplying Rational Expressions. EX 7
as long as
and
.
Step 1:
Completely factor each numerator and denominator.
Step 2:
Multiply the numerators together and the denominators together. KEEP IN FACTORED FORM
Step 3: Simplify: Divide out any factors that are common to both the numerator and denominator.
Example 3: Multiply the rational expressions. Simplify if possible.
a)
#34
c)
d)
14
6. Dividing Rational Expressions. EX 8
as long as
,
.
Dividing one rational expression by another, multiply the first by the reciprocal of the second. Simplify.
 Dividing by 2 is the same as multiplying by ___________.
 Dividing by 2x is the same as multiplying by __________.
 Dividing by
is the same as multiplying by __________.
Example 4: Divide the rational expressions.
a)
b)
c)
15
Section 6.2 - Adding and Subtracting Rational Expressions
Objectives:
 Add or subtract rational expressions with common denominators.
 Identify the least common denominator of two or more rational expressions.
 Add or subtract rational expressions with unlike denominators.
As with numerical fractions, the procedure used to add/subtract two rational expressions depends upon
whether the expressions have like or unlike denominators.

Adding/Subtracting Rational expressions with Common Denominators. EX 1
 Add or subtract their numerators and place the result over the common denominator. Simplify
when possible.
Example 1: Add or subtract. Simplify if possible.
Now suppose we wanted to add:
The process is the same with rational expressions.
 We first find the ___________________________ between the two terms.


Multiply ___________ term by the appropriate factor so that each term will have the same
________________________.
Identifying the Least Common Denominator of Rational Expressions. EX 2
STEP 1: Factor each denominator completely.
STEP 2: The LCD is the product of all unique factors each raised to a power equal to the greatest
number of times that the factor appears in any factored denominator.
Example 2: Find the LCD of the rational expressions in each list.
a)
b)
c)
d)
16

Adding/Subtracting Rational expressions with UNLIKE Denominators. EX 3, EX 4, EX 5
Example 3: Perform the indicated operation.
a)
#33.
#42.
# 102.
#103.
STEP 1: Find the LCD of the rational expressions.
STEP 2: Write each rational expression as an equivalent rational expression whose denominator is
the LCD found in STEP 1.
STEP 3: Add or subtract numerators and write the result over the common denominator.
STEP 4: Simplify the resulting rational expression.
17
Section 6.4 – Dividing Polynomials: Long Division and Synthetic Division
Objectives:
 Divide a polynomial by a monomial.
 Divide by a polynomial.
 Use synthetic division to divide a polynomial by a binomial.
1. Dividing a polynomial by a monomial. EX 1, EX 2
Example 1: Divide
a)
b)
Check: See that (quotient)(divisor)=dividend
2. Dividing a Polynomial by a Binomial (common factors)
Example 2: Divide.
3. Dividing by a Polynomial Using Long Division. EX 3 – EX 7

Process is similar to long division of real numbers.
Quick Review:

Tip: Use a place marker of 0 for any missing terms.
Example 3:
.
18
# 17.
4. Dividing by a Binomial using Synthetic Division. EX 8, EX 9
 The divisor must be in the form
; leading coefficient is 1; degree is 1.
 Use a place marker of 0 for any missing terms.
Example 4: Divide using synthetic division.
a)
b)
c)
19
Section 6.5 – Solving Equations Containing Rational Expressions
Objective: Solving equations containing rational expressions.
Solving linear equations by clearing fractions was covered in Unit I. Solving rational equations with a
variable in the denominator is an extension of the process.
Check:
1. Review :
2. Solving Equations Containing Rational Expressions. EX 1 – EX 6
We will be using the same process to solve equations:
Example 1:
Domain:
Check:
LCD:
Example 2:
Domain:
Check :
LCD:
Step 1:
Find the LCD.
Step 2:
Multiply both sides of the equation by the LCD of ALL rational expressions in the equation.
Step 3:
Simplify both sides.
Step4:
Solve the equation
Step 5:
Check the solution in the original equation.
20
#11
Domain:
LCD:
#21
Domain:
LCD:
21
Section 6.6 – Rational Equations and Problem Solving
Objectives: Solve problems by writing equations containing rational expressions.
1. Solving Equations with Rational Expressions for a Specified Variable. EX 1 – EX 5
Example 1: Greg Guillot can paint a room alone in 5 hours. His sister Maddie can do the same job alone
in 3 hours. How long would it take them to paint the room if they worked together?
Understand: The key idea here is the relationship between time (in hours) it takes to complete the job
and the part of the job completed in one unit of time (1 hour). So if Greg can paint the room
in 5 hours, he can paint 1/5 of the room in 1 hour. Similarly, Maddie can paint the room in 3
hours, so she can paint 1/3 of the room in 1 hour.
Let t = time it takes Greg and Maddie to paint the room together.
Hours to
complete the job
Part of the job
completed in 1 hour
GREG ALONE
MADDIE ALONE
TOGETHER
Translate: We use the equation
to solve work/rate problems.
Solve:
Interpret:
Example 2: Elissa Juarez can clean the animal cages at the animal shelter where she volunteers in 3
hours. Bill Stiles can do the same job in 2 hours. How long would it take them to clean
cages if they work together?
Understand: If Elissa can clean the cages in 3 hours, she can complete ______ of the job in 1 hour. If Bill
can clean the cages in 2 hours, he complete ______ of the job in 1 hour.
Hours to
complete the job
Part of the job
completed in 1 hour
ELISSA ALONE
BILL ALONE
TOGETHER
Translate:
Solve:
Interpret:
22
#28. Alan Cantrell can word process a research paper in 6 hours. With Steve Isaac’s help, the paper can
be processed in 4 hours. Find how long it takes Steve to word process the paper alone.
Example 4: The speed of a Ranger boat in still water is 32 mph. If the boat travels 72 miles upstream in
the same amount of time that it takes to travel 120 miles downstream, find the current of the stream.
Understand:
Distance
Rate
Time
UPSTREAM
DOWNSTREAM
Translate:
#31. Mattie Evans drove 150 miles in the same amount of time that it took a turbopropeller plane to
travel 600 miles. The speed of the plane was 150 mph faster than the speed of the car. Find the speed
of the plane.
Understand:
Distance
Rate
Time
CAR
PLANE
23
#33. The speed of Lazy River’s current is 5 mph. If a boat travels 20 miles downstream in the same
amount of time that it takes to travel 10 miles upstream, find the speed of the boat in still water.
Distance
Rate
Time
UPSTREAM
DOWNSTREAM
24