Lesson 6.1 – Polynomial Functions
A polynomial in one variable with constant coefficients is given by
Polynomials are expressions containing “variables and exponents.” Each piece of the
polynomial, each part that is being added, is called a "term". Polynomial terms have variables
which are raised to whole-number exponents (or else the terms are just plain numbers); there are
no square roots of variables, no fractional powers, and no variables in the denominator of any
fractions. Here are some examples:
This is NOT
a polynomial term...
...because the
variable has a
negative exponent.
1
𝑥2
This is NOT
a polynomial term...
...because the
variable is in the
denominator.
√𝑥
This is NOT
a polynomial term...
...because the
variable is inside a
radical.
This IS a polynomial
term...
...because it obeys all
the rules.
6x
–2
4x2
Example 1 – Determine which of the following expressions is a polynomial. If it is not, explain
why.
a) 𝑥 3 − √2 + 1
1
b) 2𝑥 2 − 𝑥 + 3
2
c) −3𝑥 2 +
1
𝑥
d) 5−2 𝑥 + 3𝑥 4
e)
1
3
𝑥 + 2 √𝑥 − 𝑥 3
f) 6𝑥 3 + 3𝑥 2 − 𝑥 −1
Degree of a Polynomial: _______________________________________________________
Standard Form of a Polynomial: _________________________________________________
Degree Name Using
Degree
0
Constant
1
Linear
2
Quadratic
3
Cubic
4
Quartic
5
Quintic
Number
of Terms
1
2
3
4
Name Using Number
of Terms
Monomial
Binomial
Trinomial
Polynomial of 4 terms
Example 2 – Write each polynomial in standard form. Then classify it by degree and number of
terms.
a) −7𝑥 + 5𝑥 4
b) 4𝑥 − 6𝑥 + 5
c) 3𝑥 3 + 𝑥 2 − 4𝑥 + 2𝑥 3
Example 3 – Fill in the table with the correct information:
Polynomial
2𝑥 + 3
1
3𝑥 3 + 4𝑥 − 5
2𝑥 2 − 7𝑥
6𝑥 5 − 𝑥 3 + 8𝑥 2 − 1
9𝑥 4 + 3𝑥 2 + 2𝑥 − 11
Degree
Name Using
Degree
Number of
Terms
Name Using
Number of
Terms
Example 4 – Use a graphing calculator to determine whether a linear model, a quadratic model,
or a cubic model best fits the data in the table. Then find the model.
a)
b)
x
0
2
4
6
8
y
2.8
5
6
5.5
4
Example 5 – Simplify. Write the answer in standard form:
a) (2𝑥 3 + 3𝑥 2 − 4) + (3 − 5𝑥 3 + 6𝑥 2 − 𝑥 4 )
b) (9𝑥 2 + 3𝑥 5 + 2𝑥 − 12) − (6𝑥 5 − 𝑥 3 + 8𝑥 2 − 1)
c) (3𝑥 − 2)2 (−4𝑥 + 1)
d) (2𝑎 − 3) (4𝑎 + 6) (−𝑎 − 1)
x
0
5
10
15
20
y
10.1
2.8
8.1
16
17.8
Section 6.1 - Pages 303 – 304 #’s 2-12 evens, 22, 31, 34, 44, 48, 50, and 58.
Write each polynomial in standard form. Then classify it by degree and by number of terms.
2) 5 − 3𝑥
4) – 𝑥 3 + 𝑥 4 + 𝑥
6) 5𝑎2 + 3𝑎3 + 1
8) 3 + 12𝑥 4
10) 7𝑥 3 − 10𝑥 3 + 𝑥 3
12) 𝑥 3 − 𝑥 4 + 2𝑥 3
Find a cubic model for the function. Then use your model to estimate the value of y when x = 17.
22)
x 0 3 5 6 9 11 12 14 16 18 20
y 42 31 26 21 17 15 19 22 28 30 29
31) The diagram below shows a cologne bottle that consists of a cylindrical base and a hemispherical top.
a) Write an expression for the cylinder’s volume it its height is 10 cm.
b) Write an expression for the volume of the hemispherical top.
c) Write a polynomial to represent the total volume.
Simplify. Classify each result by degree number of terms.
34) (−8𝑑 3 − 7) + (−𝑑3 − 6)
44) (−3𝑥 3 + 7𝑥 2 − 8) − (−5𝑥 3 + 9𝑥 2 − 8𝑥 + 19)
Find each product. Classify the result by degree and number of terms.
48) (2𝑎 − 5)(𝑎2 − 1)
50) (𝑥 − 2)3
58) (𝑠 + 𝑡)(𝑠 − 𝑡)(𝑠 + 𝑡)(𝑠 − 𝑡)