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 ?

A polynomial function is a function of the form:

on

nn

n axaxaxaxf 1

11

All of these coefficients are real numbers

n must be a positive integer

Remember integers are … –2, -1, 0, 1, 2 … (no decimals or fractions) so positive integers would be 0, 1, 2 …

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

x 0

2

1

xx

Not a polynomial because of the square root since the power is NOT an integer

xxxf 42

Determine which of the following are polynomial functions. If the function is a polynomial, state its degree.

A polynomial of degree 4.

2xg

12 xxh

23x

xxF

A polynomial of degree 0.

We can write in an x0 since this = 1.

Not a polynomial because of the x in the denominator since the power is negative 11 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

LinearFunction

Degree = 1

Max. Zeros: 1

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x2 + 3x + 2

QuadraticFunction

Degree = 2

Max. Zeros: 2

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x3 + 4x2 + 2

CubicFunction

Degree = 3

Max. Zeros: 3

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x4 + 4x3 – 2x – 1

QuarticFunction

Degree = 4

Max. Zeros: 4

LEFT RIGHTand

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.

left hand behaviour: falls right hand

behaviour: falls

turning points in the middle

Odd degree polynomials fall on the left and rise on the right hand sides of the graph (like x3) if the coefficient is positive.

left hand behaviour: falls

right hand behaviour: rises

turning Points in the middle

Odd degree polynomials rise on the left and fall on the right hand sides of the graph (like x3) if the coefficient is negative.

left hand behaviour: rises

right hand behaviour: falls

turning points in the middle

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).

doesn’t mean it has that many turning points but that’s the most it can have

3019153 234 xxxxxf

Let’s determine left and right hand behaviour for the graph of the function:

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 and y intercepts would be useful and we know how to find those. To find the y intercept we put 0 in for x.

3019153 234 xxxxxf

30300190150300 234 f

(0,30)

To find the x intercept we put 0 in for y.

51320 xxxx

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)