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Algebra 2 Honors
Chapter 2 Notes: Polynomials and Polynomial Functions
Section 2.1: Use Properties of Exponents
Evaluate each expression
(34 )2
3
𝑎2
( −3 )
𝑏
5 3
( )
8
(−2)−3 (−2)9
(−𝑦 2 )5 𝑦 2 𝑦 −12
𝑟𝑠
(𝑟𝑠 −1 )3
More Challenge:
𝑎2 𝑏
(𝑎−3𝑏2)
−1
2𝑚𝑛2 (3𝑚2 𝑛)2
12𝑚3 𝑛4
2
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Section: 2.3 Add, Subtract, and Multiply Polynomials
Examples:
(𝒙𝟒 + 𝟐𝒙𝟑 + 𝟖) + (𝟐𝒙𝟒 − 𝟗)
(𝒙 + 𝟐)(𝟓𝒙𝟐 + 𝟑𝒙 − 𝟏)
(𝟗𝒙𝟒 − 𝟏𝟐𝒙𝟑 + 𝒙𝟐 − 𝟖) − (𝟑𝒙𝟒 − 𝟏𝟐𝒙𝟑 − 𝒙)
(𝒙 − 𝟐)(𝒙 − 𝟏)(𝒙 + 𝟑)
Special Product Patterns
1) Sum and Difference
Example
( x  4)( x  4)  ______________________
(a  b)(a  b)  a 2  b2
2) Square of a Binomial
(a  b)2  a 2  2ab  b2
( y  3)2  ____________________________________
(a  b)2  a 2  2ab  b2
(3z 2  5)2  ___________________________________
3) Cube of a Binomial
(a  b)3  a3  3a 2b  3ab2  b3
(ab  2)3
(a  b)3  a3  3a 2b  3ab2  b3
( x  2)3  _______________________________
( p  3)3  _______________________________
Examples:
(3 x  2)(3 x  2)
(5a  2)2
(ab  2)3
4
5
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Section 2.2: Evaluate and Graph Polynomial Functions
Polynomial Function: A function where a variable x is raised to a nonnegative integer power.
The domain of any polynomial is the set of all real numbers.
n is the degree of a polynomial is the highest power of x in the polynomial. The coefficient of the highest
degree term 𝑎𝑛 is called the leading coefficient. The term in a polynomial with no variable 𝑎0 is called the
constant term and is a coefficient of itself.
Example:
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What are examples of functions that are not polynomial functions?
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End Behavior:
There are four scenarios:
Applying these principles to polynomials in standard form f ( x)  an x n  an 1 x n 1  an  2 x n 2  ...  a2 x 2  a1 x  a0 .
The end behavior of the polynomial depends on the degree, n, of the polynomial.
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Section: 2.4 Factor and Solve Polynomial Equations
Recall how to Factor Quadratic Equations…
Factor Polynomial in Quadratic Form:
𝟐𝟓𝒙𝟒 − 𝟑𝟔
Factor by grouping:
𝑥 2 𝑦 2 − 3𝑥 2 − 4𝑦 2 + 12
𝒂𝟐 𝒃𝟐 − 𝟖𝒂𝒃𝟑 + 𝟏𝟔𝒃𝟒
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Solving Polynomial Equations by Factoring:
Solve: 2𝑥 5 + 24𝑥 = 14𝑥 3
You are building a rectangular bin to hold mulch for your garden. The bin will hold 162 𝑓𝑡 3 of
mulch. The dimensions of the bin are 𝑥 𝑓𝑡. 𝑏𝑦 5𝑥 − 6 𝑓𝑡. 𝑏𝑦 5𝑥 − 9 𝑓𝑡. How tall will the bin
be?
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Graphing Polynomials in Factored Form
Sketch the graph:
𝑔(𝑥) = −(𝑥 − 1)3 (𝑥 + 2)(𝑥 − 3)
𝑓(𝑥) = −𝑥 5 + 𝑥
𝑓(𝑥) = (𝑥 + 3)2 (𝑥 + 1)(𝑥 − 2)3
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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
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Let
f ( x)  3 x 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? _________________________________________
The remainder theorem says that when a polynomial 𝑓(𝑥) is divided by a linear (1st degree)
polynomial, 𝑥 − 𝑐 and you solve it for x so it is in the form 𝑥 = 𝑐, if you evaluate 𝑓(𝑐), the
answer will be your remainder.
Example:
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The factor theorem is an extension of the remainder theorem. Recall that the remainder
theorem states that if you evaluate a polynomial with the c-value found by setting the linear
𝑥 + 𝑐 = 0, the result will be the remainder of the polynomial had we done either long or
synthetic division. The factor theorem extends that by saying if the remainder theorem
results in 0, the linear 𝑥 + 𝑐 must be a factor of the polynomial.
Example:
𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑤ℎ𝑒𝑡ℎ𝑒𝑟:
𝑎) 𝑥 + 1 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑓(𝑥) = 𝑥 4 − 5𝑥 2 + 6𝑥 − 1
𝑏) 𝑥 − 2 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 4
The remainder is ____. This means that 𝑥 − 2 is a factor
of 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 4Therefore you can write the
result as:
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Example: How many zeros does the polynomial have?
𝑓(𝑥) = (𝑥 − 3)(𝑥 − 1)2 (𝑥 + 2)3
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.
𝑓(𝑥) = 𝑥 5 − 4𝑥 4 − 7𝑥 3 + 14𝑥 2 − 44𝑥 + 120 if 2, 5, and -3 are factors. Find the other zeros of the function.
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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?
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2.6 Finding Rational Zeros
We call the list of all
𝑝
𝑞
“possible” or “potential” rational zeros.
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
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Find the all real zeros of: f ( x)  15x 4  68x3  7 x 2  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?
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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.
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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. [2nd][TRACE][zero]
2) Find Maximum and Minimum Points of a Polynomial Function. [2nd][TRACE][maximum or minimum]
3) Find a Polynomial Model that fits a given set of data. (Cubic, Quartic Regression) and make predictions.
[STAT, EDIT, input data in 𝐿1 and 𝐿2 , STAT, CALC, CubicReg or QuartReg, VARS, Y-VARS, Function 𝑌1 ]

The graph of a function has ups and down or peaks and valleys. A peak is known as a maximum (plural maxima) and a valley is termed as a minimum (plural - minima) of given function. There may be more
than one maximum or minimum in a function. The maxima and minima are collectively known as
extrema (whose singular is extremum) that are said to be the largest and smallest values undertaken
by given function at some point either in certain neighborhood (relative or local extrema) or over the
domain of the function (absolute or global extrema).
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
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
?
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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
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An open box is to be made from a rectangular piece of cardboard that is 12 by 6 feet by cutting out squares of
side length x ft from each corner and folding up the sides.
a) Express the volume of the box v ( x ) as a function of the size x cut out at each corner.
b) Use your calculator to determine what size square can be cut from the corners to give a box with a
volume of 40 cubic inches.
c) Use your calculator to approximate the value of x which will maximize the volume of the box.
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Polynomial Inequalities with degree two or more.
1. Set one side of the inequality equal to zero.
2. Temporarily convert the inequality to an equation.
3. Solve the equation for x . If the equation is a rational inequality, also determine the values of x where the
expression is undefined (where the denominator equals zero). These are the partition values.
4. Plot these points on a number line, dividing the number line into intervals.
5. Choose a convenient test point in each interval. Only one test point per interval is needed.
6. Evaluate the polynomial at these test points and note whether they are positive or negative.
7. If the inequality in step 1 reads  0 , select the intervals where the test points are positive. If the inequality
in step 1 reads  0 , select the intervals where the test points are negative.
Quadratic Inequalities
Example 1.
Solve.
a) x 2  2 x  15  0
b) x 2  2 x  15  0
c) 3x 2  11x  1  5
More Polynomial Inequalities
Example 2.
Solve.
a) ( x  5)(3x  4)( x  2)  0
d) x3  3x 2  16 x  48
b) 16 x  x3  0
c) ( x  5)2 ( x  1)  0
e)  x 2 (2 x  3)2  0