Chapter 5: Polynomials Chapter 5: Polynomials and Polynomial Functionsand Polynomial Functions5.1: Polynomial Functions
DefinitionsDefinitionsA monomial is a real number, a
variable, or a product of a real number and one or more variables with whole number exponents.◦Examples:
The degree of a monomial in one variable is the exponent of the variable.
2 35, , 3 , 4x wy x
DefinitionsDefinitionsA polynomial is a monomial or a
sum of monomials.◦Example:
The degree of a polynomial in one variable is the greatest degree among its monomial terms.◦Example:
23 2 5xy x
24 7x x
DefinitionsDefinitionsA polynomial function is a
polynomial of the variable x.◦A polynomial function has
distinguishing “behaviors” The algebraic form tells us about the
graph The graph tells us about the algebraic
form
DefinitionsDefinitionsThe standard form of a polynomial function arranges the terms by degree in descending order◦Example:
3 2( ) 4 3 5 2P x x x x
DefinitionsDefinitionsPolynomials are classified by
degree and number of terms.◦Polynomials of degrees zero through
five have specific names and polynomials with one through three terms also have specific names.
Degree
Name
0 Constant
1 Linear
2 Quadratic
3 Cubic
4 Quartic
5 Quintic
Number ofTerms
Name
1 Monomial
2 Binomial
3 Trinomial
4+ Polynomial with ___ terms
ExampleExampleWrite each polynomial in standard form. Then classify it by degree and by number of terms.
23 9 5x x
ExampleExampleWrite each polynomial in standard form. Then classify it by degree and by number of terms.
5 23 4 2 10x x
Polynomial BehaviorPolynomial BehaviorThe degree of a polynomial
function ◦Affects the shape of its graph◦Determines the number of turning points (places where the graph changes direction)
◦Affects the end behavior (the directions of the graph to the far left and to the far right)
Polynomial BehaviorPolynomial BehaviorThe graph of a polynomial function
of degree n has at most n – 1 turning points.◦Odd Degree = even number of turning
points◦Even Degree = odd number of turning
pointsThink about this:
◦If a polynomial has degree 2, how many turning points can it have?
◦If a polynomial has degree 3, how many turning points can it have?
Polynomial BehaviorPolynomial Behavior
End behavior is determined by the leading term nax
Polynomial Behavior Polynomial Behavior ExamplesExamples
4 34 6y x x x
2 2y x x
3y x
3 2y x x
ExampleExampleDetermine the end behavior of the graph of each polynomial function.34 3y x x
ExampleExampleDetermine the end behavior of the graph of each polynomial function.4 3 22 8 8 2y x x x
Increasing and DecreasingIncreasing and DecreasingRemember: We read from left to
right!
A function is increasing when the y-values increase as the x-values increase
A function is decreasing when the y-values decrease as the x-values increase
Example: Identify the parts Example: Identify the parts of the graph that are of the graph that are increasing or decreasingincreasing or decreasing
Example: Identify the parts Example: Identify the parts of the graph that are of the graph that are increasing or decreasingincreasing or decreasing
HomeworkHomeworkP285 #8 – 31all