lecture 8 section 3.2 polynomial equations

MATH 107

Section 3.2

Polynomial Functions

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Definitions

A polynomial function of degree n is a function of the form

where n is a nonnegative integer and the coefficients an, an–1, …, a2, a1, a0 are real numbers with an ≠ 0.


Definitions

A constant function f (x) = a, (a ≠ 0) which may be written as f (x) = ax0, is a polynomial of degree 0.

The term anxn is called the leading term.

The number an is called the leading coefficient, and a0 is the constant term.


Definitions

Degree Name

0 Zero function: f(x)=0

1 linear

2 quadratic

3 cubic

4 quartic

5 quintic


COMMON PROPERTIES OFPOLYNOMIAL FUNCTIONS

1. The domain of a polynomial function is the set of all real numbers.


2. The graph of a polynomial function is a continuous curve.


3. The graph of a polynomial function is a smooth curve.


EXAMPLE 1 Polynomial Functions

State which functions are polynomial functions. For each polynomial function, find its degree, the leading term, and the leading coefficient.

f (x) = 5x4 – 2x + 7

Solution


END BEHAVIOR OF POLYNOMIAL FUNCTIONS

Case 1

n Evena > 0

The graph rises to the left and right, similar to y = x2.



Case 2

n Evena < 0

The graph falls to the left and right, similar to y = –x2.



Case 3

n Odda > 0

The graph rises to the right and falls to the left, similar to y = x3.



Case 4

n Odda < 0

The graph rises to the left and falls to the right, similar to y = –x3.


EXAMPLE 2Understanding the End Behavior of a Polynomial Function

Letfunction of degree 3. Show that

be a polynomial 32P x x

when |x| is very large.Solution

When |x| is very large are

close to 0.

Therefore,


THE LEADING-TERM TEST

Its leading term is anxn.

The behavior of the graph of f as x → ∞ or as x → –∞ is similar to one of the following four graphs and is described as shown in each case.

The middle portion of each graph, indicated by the dashed lines, is not determined by this test.

Let 11 1 0... 0n n

nn nf x a x ax ax aa

be a polynomial function.


Case 1

n Evenan > 0



Case 2

n Evenan < 0



Case 3

n Oddan > 0



Case 4

n Oddan < 0



EXAMPLE 3 Using the Leading-Term Test

Use the leading-term test to determine the end behavior of the graph of

SolutionHere n = 3 (odd) and an = –2 < 0. Thus, Case 4 applies. The graph of f (x) rises to the left and falls to the right. This behavior is described as y ∞ as x –∞ and y –∞ as x ∞.


REAL ZEROS OF POLYNOMIAL FUNCTIONS

1. c is a zero of f .2. c is a solution (or root) of the equation

f (x) = 0.3. c is an x-intercept of the graph of f . The

point (c, 0) is on the graph of f .

If f is a polynomial function and c is a real number, then the following statements are equivalent.


EXAMPLE 4 Finding the Zeros of a Polynomial Function

Find all zeros of each polynomial function.

3 2

3 2

a. 2 2

b. 2 2

f x x x x

g x x x x

SolutionFactor f (x) and then solve f (x) = 0.


REAL ZEROS OF POLYNOMIAL FUNCTIONS

A polynomial function of degree n with real coefficients has, at most, n real zeros.


EXAMPLE 6 Finding the Number of Real Zeros

Find the number of distinct real zeros of the following polynomial functions of degree 3.

Solution

22

a. 1 2 3

b. 1 1 c. 3 1

f x x x x

g x x x h x x x


MULTIPLICITY OF A ZERO

If c is a zero of a polynomial function f (x) and the corresponding factor (x – c) occurs exactly m times when f (x) is factored, then c is called a zero of multiplicity m.

m Behavior of f at x=c

Odd Crosses

Even touches


ODD MULTIPLICITY OF A ZERO


EVEN MULTIPLICITY OF A ZERO


EXAMPLE 7 Finding the Zeros and Their Multiplicity

Find the zeros of the polynomial function f (x) = x2(x + 1)(x – 2), and give the multiplicity of each zero.

Solution


TURNING POINTS

A local (or relative) maximum value of f is higher than any nearby point on the graph.

A local (or relative) minimum value of f is lower than any nearby point on the graph.

The graph points corresponding to the local maximum and local minimum values are called turning points. At each turning point the graph changes from increasing to decreasing or vice versa.


TURNING POINTS

The graph of f has turning points at (–1, 12) and at (2, –15).


NUMBER OF TURNING POINTS

If f (x) is a polynomial of degree n, then the graph of f has, at most, (n – 1) turning points.


EXAMPLE 8 Finding the Number of Turning Points

Use a graphing calculator and the window –10 x 10; –30 y 30 to find the number of turning points of the graph of each polynomial.

4 2

3 2

3 2

a. 7 18

b. 12

c. 3 3 1

f x x x

g x x x x

h x x x x



Solution

f has three total turning points; two local minimum and one local maximum.



Solution continued

g has two total turning points; one local maximum and one local minimum.



Solution continued

h has no turning points, it is increasing on the interval (–∞, ∞).


GRAPHING A POLYNOMIAL FUNCTION

Step 1 Determine the end behavior. Apply the leading-term test.

Step 2 Find the zeros of the polynomial function. Set f (x) = 0 and solve. The zeros give the x-intercepts.

Step 3 Find the y-intercept by computing f (0).


Step 4 Draw the graph. Use the multiplicities of each zero to decide whether the graph crosses the x-axis.

Use the fact that the number of turning points is less than the degree of the polynomial to check whether the graph is drawn correctly.


EXAMPLE 9 Graphing a Polynomial Function

Sketch the graph of 3 24 4 16.f x x x x Solution

Step 1 Determine end behavior. Degree = 3 Leading coefficient = –1End behavior shown in sketch.



Solution continued

Step 2 Find the zeros by setting f (x) = 0.

Each zero has multiplicity 1, the graph crosses the x-axis at each zero.



Solution continued

Step 3 Find the y-intercept by computing f (0).The y-intercept is f (0) = 16. The graph passes through (0, 16).



Solution continued

Step 4 Draw the graph.

The number of turning points is 2, which is less than 3, the degree of f.

lecture 8 section 3.2 polynomial equations

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