Algebra II
6-1 Polynomial Functions
A polynomial is a monomial or the sum of monomials. x can be raised to any nonnegative
integer exponent, and can have any real number coefficient.
Examples: x 2 + 2, 5, 3x 4 + 2x 3 − 7,
1 3
x + 2x, x 3 + 2x 2
2
1
+ 2x,
x +2
x3
The degree of a term is the exponent on the variable. The degree of a polynomial is the largest
degree of any term of the polynomial.
Non-examples: x −2 + 3,
Standard form of a polynomial means that the terms are in descending order by degree (the
biggest exponent is first, followed by the next largest, etc.)
Classifying by Degree
Degree
Name
Polynomial Example
0
constant
5
1
linear
5x + 2
2
quadratic
5x − 3x −12
3
cubic
−6x 3 − 3x − 7
4
quartic
8x 4 + 2
5
quintic
7x 5 − x 4 + 2x 3 + x 2 + 2x − 7
Classifying by Number of Terms
# of Terms
Name
1
2
2
Polynomial Example
monomial
binomial
3
trinomial
More than 3
polynomial of # terms
7x
2
x – 4x
6
4x + 3x4 – 9
Ex. 1: Write in standard form. Then classify by the degree and number of terms.
a) -4p + 3p + 2p2
2p2 – p
quadratic binomial
b) 3 + 12x4
12x4 + 3
quartic binomial
c) (-8d3 – 7) + (-d3 – 6)
-9d3 – 13
cubic binomial
d)
(−12x
3
+ 5x − 23) − ( 4 x 4 + 31− 9x 3 )
−4 x 4 − 3x 3 + 5x − 54
quartic polynomial (of 4 terms)
e) x(2x)(4x + 1)
2x2(4x + 1)
8x3 + 2x2
cubic binomial
f) (x – 2)3 (Does NOT = x3 – 23!!!!!!)
(x – 2)(x – 2)(x – 2)
(x2 – 4x + 4)(x – 2)
x3 – 2x2 – 4x2 + 8x + 4x – 8
x3 – 6x2 + 12x – 8
cubic polynomial (of 4 terms)
g) (x + 3)(x – 2)2
(x + 3)(x2 – 4x + 4)
x3 – 4x2 + 4x + 3x2 – 12x + 12
x3 – x2 – 8x + 12
cubic polynomial (of 4 terms)
Ex. 2: Find a cubic model for the following set of values: (1, 91), (10, 95), (20, 260), (30, 365)
Then use the model to find y when x = 23.
Enter the x-values in L1 and the y values in L2.
STAT  CALC  6:CubicReg
Round to 4 significant figures for each coefficient:
y = -0.03948x3 + 2.069x2 – 17.93x + 106.9
To evaluate x = 23: y = -0.03948(23)3 + 2.069(23)2 – 17.93(23) + 106.9
= -480.35316 + 1094.501 – 412.39 + 106.9
= 308.65784 ≈ 308.6