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Algebra II – Chapter 6 Day #1
Topic: Introduction to Polynomial Functions
Today, we will begin our study of polynomial functions.
Standards/Goals:
 F.IF.7.c: I can identify the degree of a polynomial function.
 I can describe the ‘end behavior’ of a polynomial.
 I can identify the intervals where a polynomial increases or decreases.
We want to introduce ourselves to polynomials and compare them to monomials.
Vocabulary:
1. Monomial – Is a real number, a variable, or a product of a real number and one or more
variables with whole number exponents.
2. Polynomial – Is a monomial or a sum of monomials.
3. Degree of a Polynomial – Is the one variable with the greatest exponent among its
monomial terms.
A polynomial is written in the following form:
f(x) = 𝑎𝑥 𝑛 + 𝑏𝑥 𝑛−1 + 𝑐𝑥 𝑛−2 + ⋯ + k , where ‘k’ is a constant, and where a, b, and c represent
coefficients. These coefficients DO NOT have to be real. We will get into the issue of this later.
EXAMPLE: Consider the following polynomial: f(x) = 6𝑥 5 − 14𝑥 4 + 2𝑥 3 − 11𝑥 2 + 12𝑥 + 9
a. What is the degree?
b. Is this written in ascending or descending order?
c. Classify each term of the polynomial.
i.
ii.
iii.
iv.
v.
vi.
d. What is the classification of the polynomial by degree? What is the classification by the
number of terms?
Examples:
Write each polynomial in standard form. What is the classification of each polynomial by degree?
By the number of terms
#1. 3𝑥 + 9𝑥 2 + 5
#2. −3𝑥 + 4𝑥 3 + 7𝑥 − 3
Now, try these:
#3. 4𝑥 − 6𝑥 2 + 𝑥 4 + 10𝑥 2 − 12
#4. 2𝑥 3 − 3𝑥 2 + 𝑥 5
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We also want to think about what the graphs of polynomials will LOOK LIKE.
Memorize these facts:
a. The graph of a polynomial function is continuous for ALL real numbers.
b. The graph of a polynomial function has NO sharp corners (like an absolute value function).
c. The graph of a polynomial function has at most (n-1) turning points. (n = the largest exponent
of the functions/degree). A turning point is a place where the graph changes direction.
d. The graph of a polynomial function increases or decreases without bound as x-approaches +∞
and as x approaches−∞.
We want to consider that last fact (part d, above) and think about the ‘end behavior’ of the graph
of a polynomial function. An easy way to MEMORIZE is by ONLY memorizing the following facts:
Consider the graphs of:
#1. P(x) = 𝑥 2
#2. P(x) = −𝑥 2
#3. P(x) = 𝑥 3
#4. P(x) = −𝑥 3
Summary:
‘n’ is even
‘n’ is odd
Lead Coefficient is positive
Up and Up
Down and Up
Lead Coefficient is Negative Down and Down Up and Down.
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Examples:
Consider the leading term of: 𝑦 = −2𝑥 2 − 3𝑥 + 3. What is the end behavior of the graph? How
many turning points will it have?
Consider the leading term of: 𝑦 = 3𝑥 4 − 2𝑥 3 + 𝑥 − 1. What is the end behavior of the following
graphs?
Do the same for: 𝑦 = 3 − 2𝑥 3 + 𝑥.
We also want to be able to describe the intervals of increasing/decreasing.
Consider the following functions: Determine the intervals in which it is increasing/decreasing.
#1.
#3.
#2.
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HOMEWORK – Chapter 6 Day #1
Write each polynomial in standard form. Then classify it by degree and by number of
terms.
1. 4x3  3 + 2x2
2. 8  x5 + 9x2  2x
3. 6x + 2x4  2
4. 6x3 +5. 3 + 24x2
5. (5s3  2s2)  (s4 + 1)
Describe the shape of the graph of each cubic function by determining the end
behavior and number of turning points.
6. y = 5x3  2x2 + 1
7. y = 5  x + 4x2
8. y = x  x2 + 10
9. y = 3x2 + 9  x3
10. y = 8x2  4x4 + 5x7  2
11. y = 20  x5
12. y = 1 + 2x + 4x3  8x4
13. y = 15  5x6 + 2x  22x3
14. y = 3x + 10 + 8x4  x2
Consider the following functions: Determine the intervals in which it is
increasing/decreasing.
#15.
#16.
#17.
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