2.4
Factor and Solve Polynomial Equations
Recall how to Factor Quadratic Equations…
2.5
Apply the Remainder and Factor Theorems
Dividing Polynomials
When you divide a polynomial f ( x) by a divisor d ( x ) , you get a quotient polynomial q( x) and
f ( x)
r ( x)
a remainder r ( x ) . We must write this as
.
 q ( x) 
d ( x)
d ( x)
Method 1) Using Long Division: Divide y 4  2 y 2  y  5 by y 2  y  1
Divide
x3  x 2  2 x  8
by
x 1
Let
f ( x)  3x 3  2 x 2  2 x  5
1. Use long division to divide f ( x) by x  2
What is the Quotient? ________ What is the Remainder? ________
2. Use Synthetic Substitution to evaluate f (2) . _________
How is f (2) related to the remainder? _____________________.
What do you notice about the other constants in the last row of the synthetic
substitution? _________________________________________
Remainder Theorem: If a polynomial
r  f (k )
f ( x ) is divided by x  k , then the remainder is
Method 2) Using Synthetic Division: Divide
binomial.
(a) x  4
(b)
f ( x)  x 3  3x 2  7 x  6
by each
x2
The remainder is ____. This means that x  2 is a factor of
f ( x)  x3  3x 2  7 x  6 Therefore you can write the result as:
Factor Theorem:
A polynomial f ( x) has a factor x  k , if and only if f (k )  0
Factoring Polynomials
Factor f ( x)  3x3  13x 2  2 x  8 given that f (4)  0
Solving Polynomials (which also means Finding the ________)
One zero of f ( x)  x3  6 x 2  3x  10 is x  5 . Find the other zeros of the function.
Using Polynomial Division in Real Life
A company that manufactures CD-ROM drives would like to increase its production. The demand function for
the drives is p  75  3x2 , where p is the price the company charges per unit when the company produces x
million units. It costs the company $25 to produce each drive.
a) Write an equation giving the company’s profit as a function of the number of CD-ROM drives it
manufactures.
b) The company currently manufactures 2 million CD-ROM drives and makes a profit of $76,000,000. At
what other level of production would the company also make $76,000,000?
2.6
The Rational Zero Theorem: If
Finding Rational Zeros
f ( x)  an x n  an 1 x n 1  an  2 x n  2  ...  a2 x 2  a1 x  a0
integer coefficients, then every rational zero of f has the following form:
factorof terma0
p

q factorof an
Find the rational zeros of
f ( x)  x3  4 x 2  11x  30 .
 List the possible rational zeros:
 Test (Verify zero using the Remainder Theorem)
 Factor
has
4
3
2
Find the all real zeros of f ( x)  15x  68x  7 x  24 x  4 .
Solving Polynomial Equations in Real Life
A rectangular column of cement is to have a volume of 20.25 ft.3 The base is to be square, with sides 3 ft. less
than half the height of the column. What should the dimensions of the column be?
A company that makes salsa wants to change the size of the cylindrical salsa cans. The radius of the new can
will be 5 cm. less than the height. The container will hold 144∏ cm3 of salsa. What are the dimensions of the
new container?
2.7
Finding All Zeros of Polynomial Function
Use Zeros to write a polynomial function
Example 1: Find all the zeros of f ( x)  x5  2 x 4  8x 2  13x  6
Example 2: Write a polynomial function f(x) of least degree that has real coefficients, a
leading coefficient of 1, and 2 and 1+i as zeros.
Example 3: Write a polynomial function f(x) of least degree that has real coefficients, a
leading coefficient of 2, and 5 and 3i are zeros.
2.8 Analyzing Graphs of Polynomial Functions -Using the Graphing Calculator
1) Approximate Zeros of a Polynomial Function
2) Find Maximum and Minimum Points of a Polynomial Function
3) Find a Polynomial Model that fits a given set of data. (Cubic, Quartic Regression) and make predictions.

Identify the zeros (x-intercepts), maximums and minimums of
f ( x)  x 3  2 x 2  5 x  1
and
f ( x)  2 x 4  5 x 3  4 x 2  6

A rectangular piece of sheet metal is 10 in. long and 10 in. wide. Squares of side length x are cut from the
corners and the remaining piece is folded to make an open top box.
a) What size square can be cut from the corners to give a box with a volume of 25 cubic inches.
b) What size square should be cut to maximize the volume of the box? What is the largest possible volume of
the box?
Use your Graphing Calculator to find the appropriate polynomial model that fits the data. Use it to make predictions.
x
f(x)
1
26
2
-4
3
-2
4
2
5
2
6
16
…….. 10
?
The table shows the average price (in thousands of dollars) of a house in the Northeastern United States for 1987 to
1995. Find a polynomial model for the data. Then predict the average price of a house in the Northeast in 2000.
x
f(x)
1987
140
1988
149
1989
159.6
1990
159
1991
155.9
1992
169
1993
162.9
1994
169
1995
180