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Polynomial Degree and Finite
Differences
Objective: To define polynomials
expressions and perform
polynomial operations
Definitions:
 Polynomial Expression: The sum of terms
containing the same variable raised to different
powers
 Polynomial Function:
 A function where all exponents are whole numbers and
the coefficients are all real numbers
 Written in the form anxn + an-1xn-1 + … + a1x1 + a0
Examples:
 Are each of the following polynomial functions?
1. f(x) = -5x5 + 3x4 – 2x + 8
Yes
2. f(x) = 5 – 2x
Yes
Yes
3.
4.
5.
6.
f(x) = -8
f(x) = x2 + 2x
f(x) = 6x2 – 5x-1
1
f(x)  7x  2x 
x
4
No
No
No
Definitions:
Terms – the number of monomials that
make up the polynomial
Constant – the number without a variable
Polynomials are named by their terms:
Monomial: a polynomial that has one term
Binomial: a polynomial that has two terms
Trinomial: a polynomial that has three terms
Polynomial: if there are more than three terms
Definitions:
 Degree of a monomial – the sum of the exponents
of the variables
 Degree of a polynomial – the degree of the term
that has the highest degree
 Leading coefficient – the coefficient of the first
term when the polynomial is in standard form
 Standard form – the degrees of the terms are
written in descending order from left to right
 Even or Odd function – determined by the degree
of the function
Example:
 f(x) = 2 – x + 5x4 – 3x2 + 2x3
 Standard form:
 f(x) = 5x4 + 2x3 – 3x2 – x + 2
 Leading coefficient:
5
 Even or odd:
 Even
 Degree of each monomial:
 4, 3, 2, 1, 0
 Degree of the polynomial:
4
Practice:
 Identify the degree of each polynomial:
 3x4 – 2x3 + 3x2 – x + 7
4
 x5 – 1
5
 0.2x – 1.5x2 + 3.2x3
3
 250 – 16x2 + 20x
2
 x + x2 – x3 + x4 – x5
5
 5x2 – 6x5 + 2x6 – 3x4 + 8
6
Practice:
Determine which of the expressions are
polynomials. For each polynomial, state its
degree and write it in standard form. If it is
not a polynomial, explain why not.
1 + x2 – x3 polynomial? yes no
Standard form:
-x3 + x2 + 1
0.2x3 + 0.5x4 + 0.6x2 polynomial: yes no
Standard form:
0.5x4 + 0.2x3 + 0.6x2
Practice:

1
x 2
x
polynomial? yes
no
 can’t have a variable in a denominator
 25
 Standard form:
polynomial? yes
no
25
  2 x2  3 x3  5  5 x
polynomial? yes
3
5
12 8
3 3 2 2 5
5
 Standard form:
x  x  x
5
3
8
12

x  3x 2  5
polynomial? yes
 Can’t have a variable inside a radical
no
no
Polynomial Operations
 To simplify polynomials:
 Combine like terms by adding or subtracting their
coefficients
 To add polynomials:
 Combine like terms by adding or subtracting their
coefficients
 To subtract polynomials:
 Remember that subtracting is the same as adding the
opposite, so change the sign on the coefficient of each
term being subtracted
 To multiply polynomials:
 Use the distributive property then combine like terms
Practice Problems:
(-x2 + 2x – 1) + (6x2 – x + 5)
5x2 + x + 4
(2x2 + 4x + 3) + (3x2 – 9)
5x2 + 4x – 6
(3x2 – 2x + 6) + (-x2 – 3)
2x2 – 2x + 3
(7x2 + x – 8) – (6x2 – 3x + 5)
7x2 + x – 8 – 6x2 + 3x – 5
x2 + 4x - 13
Practice Problems
(5x3 – 5x2 + 2x – 3) – (-x2 + 4x + 3)
5x3 – 5x2 + 2x – 3 + x2 – 4x – 3
5x3 – 4x2 – 2x – 6
x(7x2 – 2x + 1)
7x3 – 2x2 + x
(x2 + 3)(x2 + 2x – 4)
x4 + 2x3 – 4x2 + 3x2 + 6x – 12
x4 + 2x3 –x2 + 6x - 12
Practice Problems
(x + 3)(x – 1)(x + 5)
(x2 – x + 3x – 3)(x + 5)
(x2 + 2x – 3)(x + 5)
x3 + 5x2 + 2x2 + 10x – 3x – 15
x3 + 7x2 + 7x - 15