Section 3.1

Polynomial Functions and Models

Polynomial FunctionsA polynomial of degree n is a function of the form

P(x) = anxn + an-1xn-1 + ... + a1x + a0

Where an 0. The numbers a0, a1, a2, . . . , an are

called the coefficients of the polynomial. The a0 is

the constant coefficient or constant term. The number an, the coefficient of the highest power,

is the leading coefficient, and the term anxn is the

leading term.

Example of a Polynomial Function

4 3( ) 4 2 5 3P x x x x

Graphs of Polynomial Functions and Nonpolynomial

Functions

Graphs of Polynomials

• Graphs are lines– Degree 0 or 1 ex. f(x) = 3 or f(x) = x –

5

• Graphs are parabolas– Degree 2 ex. f(x) = x2 + 4x + 8

• Graphs are smooth curves– Degree greater than

2 ex. f(x) = x3

• These graphs will not have the following:– Break or hole– Corner or cusp

End Behavior of Polynomials

End Behavior- a description of what happens as x becomes large in the positive and negative direction.

End Behavior is determined by:•Term with the highest power of x •Sign of this term’s coefficient

Even- and Odd-Degree Functions

The Leading-Term Test

Finding Zeros of a Polynomial

Zero- another way of saying solution

Zeros of Polynomials• Solutions• Place where graph crosses the x-axis

(x-intercepts)• Zeros of the function

Place where f(x) = 0

X-Intercepts (Real Zeros)

• A polynomial function of degree n will have at most n x-intercepts (real zeros).

Number of Turning Points (relative maxima/minima)

The number of relative maxima/minima of the graph of a polynomial function of degree n is at most n – 1.

ex. f(x) = x4 + 3x3 – 2x2 + 1

Determine number of relative maxima/minima n – 1 = 4 – 1 = 3

Using the Graphing Calculator to Determine

Zeros

4 3 2( ) 5 21 18P x x x x x Graph the following polynomial function and determine the zeros.

Before graphing, determine the end behavior and the numberof relative maxima/minima.

In factored form:P(x) = (x + 2)(x – 1)(x – 3)²

MultiplicityIf (x-c)k, k 1, is a factor of a polynomial

function P(x) and:

K is odd– The graph crosses

the x-axis at (c, 0)

K is even– The graph is

tangent to the x-axis at (c, 0)

Multiplicity

y = (x + 2)²(x − 1)³

 Answer.  

 −2 is a root of multiplicity 2,

and 1 is a root of multiplicity 3.  

These are the 5 roots:

−2,  −2,  1,  1,  1.

Multiplicity

y = x³(x + 2)4(x − 3)5

Answer.   

0 is a root of multiplicity 3,

-2 is a root of multiplicity 4,

and 3 is a root of multiplicity 5.  

True or False?• 1.) The function must

have 1 real zero.

• 2.) The function has no real zeros.

• 3.) An odd degree polynomial function must have at least 1 real zero.

• 4.) An even degree polynomial function must have at least 1 real zero.

3 2( ) 3 2 5P x x x x

4( ) 3 5P x x

To Graph a Polynomial1. Use the leading term to determine the end behavior.

2. Find all its real zeros (x-intercepts). Set y = 0.

3. Use the x-intercepts to divide the graph into intervals and choose a test point in each interval to graph.

4. Find the y-intercept. Set x = 0.

5. Use any additional information (i.e. turning points or multiplicity) to graph the function.

The Intermediate Value Theorem

Consider a polynomial function P(x) with the points (a, P(a)) and (b, P(b)) on the function.

For any P(x) with real coefficients, suppose that for a ≠ b, P(a) and P(b) are of opposite signs. Then the function has a real zero between a and b.

The Intermediate Value Theorem

In other words, if one point is above the x-axis and the other point is below the x-axis, then because P(x) is continuous and will have to cross the x-axis to connect the two points, P(x) must have a zero somewhere between a and b.