7.1 Rational Algebraic Functions
Review Material
Be sure you have reviewed fraction arithmetic.
a. Define rational (number) and give 2
examples
b. Define polynomial and give two examples of
expressions that are polynomials and one
example of an expression that is not a
polynomial and say why not.
c. Define function and give an example of an
equation which represents a function and an
equation which does not represent a function
d. Define asymptote and sketch the graph of a
function which has one vertical asymptote (no
equation reauired).
e. Define discontinuity for a graph of a function,
and sketch the graph of a function with a
discontinuity that is not an asymptote.
1. Copy the objective.
2. Copy the definition of rational algebraic
expression and give 2 examples. Give one
example of an expression which is not a rational
(algebraic) expression and say why not.
3. Copy the definition of rational algebraic
function (also called rational function) and give
two examples.
7.2 Rational Function Graphs
1. Copy the objective.
2. Following the demonstration in the reading,
simplify the following rational function, identify
the removable discontinuity, and plot the graph
x4
y 2
x  2x  8
3. State the multiplication property of fractions
(see chapter 1).
4. Define removable discontinuity in the graph
 Give an example of an equation
of a function.
which has a graph with a removable
discontinuity and an asymptote.
5. Challenge!! Give an example of an equation
for a function which has a removable
discontinuity and no asymptote.
7.3 Special Products and Factoring
Review
Define complex conjugate and give an example.
Explain in complete English sentences how to
multiply 2 binomials. Give an example.
Explain in complete English sentences the
process of factoring a quadratic trinomial. Factor
x2+3x-18
How do you recognize a trinomial that is a
perfect square? Give 3 examples of trinomials
that are perfect squares, and one example of a
quadratic trimial which is not a perfect square.
1. Copy the objective.
2. Define conjugate binomials and give two
examples.
3. Copy the rule for the product of conjugate
binomials.
Example 1. Factor completely 9x2-4y2
Example 2. Multiply (x-3)(2x+5)(x-2)
Example 3. Factor completely 6x6-726x2
Example 4. Factor completely 5x2-22x15****(read example carefully)
Change the completely factored form
agreement on page 331. It should read:
A polynomial is completely factored when it is
written as a product of the maximum possible
number of polynomials with integer coefficients.
7.4 More Factoring and Graphing
1. Copy the objective.
2. Copy the factored forms for the sum or
difference between 2 cubes (blue box on page
336) Write a specific example for the factored
form of a difference between 2 cubes and a
specific example for the factored form of the sum
of two cubes,
3. Copy the procedure for factoring a quadratic
trinomial by splitting the middle term (blue box
p. 337)
Add Step 4: Factor the first 2 terms and the last
2 terms separately
Step 5: Undistribute the common factor.
Use this procedure to factor 5x2-7x-6
4. Define prime quadratic.
5. Copy the discriminant test for prime
quadratics (blue box p. 338) Determine which of
the following quadratics are prime.
4x2+3x-2;
3x2+7x-6; 5x2-3x+1
Example 1. Factor 8x3+729
7.5 Long Division of Polynomials
1. Copy the objective.
2. When can division be used to transform a
rational expression into a "mixed number"?
3. Following the example in the book, divide
the following rational expression.
5x 4  2 x3  6 x 2  x  4
( x  3)
Example 1: Divide x3+x-7 by x+2
7.6 Higher Degree Polynomials –
Factor Theorem
1. Copy the objective.
2. Factor the polynomial 6x3+9x2+ 10x +15
3. For any polynomial (degree 2, or 3 or 4 or 5
or….) if there is a linear factor of the form ax+
b, what is the relationship between a and the
leading coefficient in the polynomial? What is
the relationship between b and the constant term
in the polynomial? [Read page 348 carefully]
4. Copy the Factor Theorem. Explain in
complete English sentences how it helps in
factoring polynomials?
Example 1: Find the factors of x3+x2-17x+15
Example 2: Find the factors of 3x3-x2+7x+6
5. Copy the Rational Root Theorem.
6. What is the factor pattern for a polynomial
which is a sum between 2 like odd powers
(xn+bn where n is odd?
Factor x5+243
7. What is the factor pattern for a polynomial
which is the difference between 2 like odd
powers (xn-bn where n is odd?
Example 3: Factor x7- 16,384
8. What happens to these 2 rules when n is even
7.7 Multiplying and Dividing Rational
Expressions
1. Explain in CES how you multiply 2
fractions. Multiply 6/10 x 5/12
2. Explain in CES how you divide 2 fractions.
Divide 6/10 by 2/5
3. Copy the objective.
4. Write down the multiplication and division
properties of fractions.
Example: Using the example starting on the
bottom of page 356, multiply the following
x 2  3x 10 x 2  4 x  3

x 2  7x  12 x 2  3x  2
5. Copy the definition of a simplified rational
expression.
6. Copy the definition of canceling in a fraction.

x3
7. Explain why
is not equal to 3/4.
x4
7.8 Sums and Differences of Rational
Expressions

1. Explain in GES how to add or subtract
fractions with unlike denominators. Subtract 1/6
from 7/15.
2. Copy the objective.
Example: Following the example on page 365
9x  4
3x
add 2
 2
x  3x 10 x  3x  2
3. Write the four factors to keep in mind when
adding and subtracting rational expressions (p.
366)
4.Read partialfraction problem p. 369.
5x  43
5. Separate into partial fractions
2
2x  3x  35
7.9 Graphs of Rational Algebraic
Functions
1. Copy the objective.
2. Review, what does a hole look like in the
equation of a rational function?
3. Review, what does a vertical asymptote look
like in the equation of a rational function?
1
4. For the function f (x) 
, plot the points
x4
for x=3.9, 3.99. 4.1, and 4.01. What are the
values like?
5,. Define indeterminate. When is the value of

a rational expression indeterminate?
6. Copy the conclusion box on p.372.
Example: Draw the graph of
x2  4
f (x)  2
x  4 x 12
7. Review, what is the quickest way to draw the
graph of a quadratic function? Draw the graph
of x 2  8x  48

7.10 Fractional Equations and
Extraneous
 Solutions
1. Copy the objective.
2. Define fractional equation and give two
examples. Give an example of an equation that
contains fractions but is not a fractional equation.
3. Define extraneous solution ( review).
Fractional equations frequently result in
extraneous solutions.
4. Copy the solution technique in the box on
p.379
Example Solve the following equation
x
7
14

 2
x  2 x  5 x  3x 10

7.11 Variation Functions
1. Copy the objective.
2. Define variation function.
3. Define constant of proportionality
4. What are the two types of variation function
discussed here? Give a specific – with numbersexample of each.
5. Copy the definition box on page 383.
6. Copy the equations box on page 384
7. Look carefully at the graphs on the top of
page 385. Using these as a model, sketch the
graph of the variation function f (x)  kx 3 and
k
f (x)  3
x
8. Copy the properties box on p.387.

9. If y is inversely proportional to the cube of x,
what happens to y when I multiply x by 5 (show

algebraically)?
10. If y is directly proportional to the 4th power
of x, what happens to y when I multiply x by 3
(show algebraically)?
Example 1: Look at problem 2 on page 393
Example 2: Look at problem 8 on page 395.
10. Copy the geometrical objects properties box
on page 391.