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POLYNOMIAL FUNCTIONS
A POLYNOMIAL is a monomial
or a sum of monomials.
A POLYNOMIAL IN ONE
VARIABLE is a polynomial that
contains only one variable.
Example: 5x2 + 3x - 7
POLYNOMIAL FUNCTIONS
The DEGREE of a polynomial in one variable is
the greatest exponent of its variable.
A LEADING COEFFICIENT is the coefficient
of the term with the highest degree.
What is the degree and leading
coefficient of 3x5 – 3x + 2 ?
Remember integers are … –2, -1, 0, 1, 2 … (no decimals
or fractions) so positive integers would be 0, 1, 2 …
A polynomial function is a function of the form:
n must be a positive integer
f x   an x  an 1 x
n
n 1
   a1 x  ao
All of these coefficients are real numbers
The degree of the polynomial is the largest
power on any x term in the polynomial.
Polynomial Functions
• Exponents must be non-negative
integer exponents
• Can not have variables in the
denominator
• Can not have radicals
– Example: square root or cube root
– These are actually fractional exponents
Determine which of the following are polynomial
functions. If the function is a polynomial, state its
degree.
f x   2 x  x
4
g x   2 x 0
h x   2 x  1
3
2
F x    x
x
A polynomial of degree 4.
We can write in an x0 since this = 1.
A polynomial of degree 0.
Not a polynomial because of the
square root since the power is NOT
1
an integer
x  x2
Not a polynomial because of the x in
the denominator since the power is
1
1
negative
x
x
Graphs of polynomials are smooth and continuous.
No sharp corners or cusps No gaps or holes, can be drawn
without lifting pencil from paper
This IS the graph
of a polynomial
This IS NOT the graph
of a polynomial
POLYNOMIAL FUNCTIONS
A polynomial equation used to represent a
function is called a POLYNOMIAL FUNCTION.
Polynomial functions with a degree of 1 are called
LINEAR POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 2 are called
QUADRATIC POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 3 are called
CUBIC POLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = 3
Constant
Function
Degree = 0
Max. Zeros: 0
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x + 2
Linear
Function
Degree = 1
Max. Zeros: 1
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
Quadratic
Function
Degree = 2
Max. Zeros: 2
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x3 + 4x2 + 2
Cubic
Function
Degree = 3
Max. Zeros: 3
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x4 + 4x3 – 2x – 1
Quartic
Function
Degree = 4
Max. Zeros: 4
LEFT
and
RIGHT
HAND BEHAVIOUR OF A GRAPH
The degree of the polynomial along with the sign of the
coefficient of the term with the highest power will tell us
about the left and right hand behaviour of a graph.
Even degree polynomials rise on both the left and
right hand sides of the graph (like x2) if the coefficient
is positive. The additional terms may cause the
graph to have some turns near the center but will
always have the same left and right hand behaviour
determined by the highest powered term.
left hand
behaviour: rises
right hand
behaviour: rises
Even degree polynomials fall on both the left and
right hand sides of the graph (like - x2) if the
coefficient is negative.
turning points
in the middle
left hand
behaviour: falls
right hand
behaviour: falls
Odd degree polynomials fall on the left and rise on
the right hand sides of the graph (like x3) if the
coefficient is positive.
turning Points
in the middle
left hand
behaviour: falls
right hand
behaviour: rises
Odd degree polynomials rise on the left and fall on
the right hand sides of the graph (like x3) if the
coefficient is negative.
turning points
in the middle
left hand
behaviour: rises
right hand
behaviour: falls
A polynomial of degree n can have at most n-1 turning
points (so whatever the degree is, subtract 1 to get
the most times the graph could turn).
Let’s determine left and right hand behaviour for the
graph of the function:
doesn’t mean it has that many
4
3 turning2points but that’s the
f x  x  3x most
15xit can
 19
x

30
have

degree is 4 which is even and the coefficient is positive so the
graph will look like x2 looks off to the left and off to the right.
The graph can
have at most 3
turning points
How do we
determine
what it looks
like near the
middle?
Characteristics
• Maximum number of turns in 1 less than
the degree
• Degree
– Odd with positive leading coefficient
• Starts down and comes up
– Even with positive leading coefficient
• Starts up and comes down
• Negative leading coefficient changes
direction of starting position
x 23xx 315
xx 119xx 30
0f x x 
5
4
3
2
x and y intercepts would be useful and we know how
to find those. To find the y intercept we put 0 in for x.
f 0  0  30  150  190  30  30
4
3
2
To find the x intercept we put 0 in for y.
Finally we need a smooth
curve through the
intercepts that has the
correct left and right hand
behavior. To pass through
these points, it will have 3
turns (one less than the degree
so that’s okay)
(0,30)