Module 6 Lecture Notes - People

Module 6 Lecture Notes

MAC1105

Summer B 2019

6 Polynomial Functions

6.1 End and Zero Behavior

Note 1. A polynomial of degree 2 or more has a graph with no sharp turns or cusps.

Note 2. The domain of a polynomial function is .

Definition

The values of x for which f(x) = 0 are called the or x-intercepts of f .

Note 3. If a polynomial can be factored, we can set each factor equal to zero to find the x-intercepts

(or zeros) of the function. Recall that the x-intercepts of a function are where f(x) = 0, or y = 0.

The y-intercepts are where x = 0.

How to Find the x-Intercepts of a Polynomial Function, f , by Factoring

1. Set .

2. If the polynomial function is not in factored form, then factor the polynomial.

nr divotAPOLYNOMIAL µ4POLYNOMIAL

FUNCTION FUNCTION

C 0,0

ZEROS

µdXINTERCEPTSHEROS

y INTERCEPT

f1 3 0

3. Set each factor equal to to find the x-intercepts.

Example 1. Find the x and y-intercepts of:

g(x) = (x� 2)2(2x+ 3)

Note 4. The graphs of polynomials behave di↵erently at various x-intercepts. Sometimes, a graph

will the horizontal x-axis at the x-intercepts, and other times the graph

will or bounce o↵ the horizontal x-axis at the x-intercepts.

Definition

The number of times a given factor appears in the factored form of a polynomial is called the

.

Example 2. From the above example, g(x) = (x � 2)2(2x + 3), the factor associated to the zero

at x = 2 has multiplicity . This zero has even multiplicity. The factor associated to the zero

at x = �3

2has multiplicity . This zero has odd multiplicity.

Graphical Behavior of Polynomials at x-Intercepts (Zeros)

If a polynomial contains a factor in the form (x� h)p, the behavior near the x-intercept h is deter-

mined by the power p. We say that x = h is a zero of p. The graph of a

polynomial function will touch the x-axis at zeros with multiplicities. The

graph of a polynomial function will cross the x-axis at zeros with multi-

plicities. The sum of the multiplicities is the of the polynomial function.

2

O

THISZEROHAS MOBILE 2,44tX INTERCEPTS X 2 0 AND 2 3 0 X INTERCEPTS

2 X 3g ARE 2,0 AND 32,0Y INTERCEPT 9107 0 251210 3 C 2 2 3 4.3 121Y INTERCEPT

1510,12CROSSTOUCH

MULTIPLICITY

2

7

MULTIPLICITYEVEN

ODD

DEGREE

Example 3. The graphs below exemplify the behavior of polynomials at their zeros with di↵erent

multiplicities:

Note 5. The graph of a polynomial function of the form

f(x) = a0 + a1x+ ...+ an�1xn�1

+ anxn

will either or as x increases without bound and will either or

as x decreases without bound. This is called the

of a function.

3

9 g 9ODDMULTIPLICITY EVEN ODDMULTIPLICITYCROSSES X AXIS MULTIPLICITY CROSSESAT THE ZERO GRAPHTOUCHES

X AXISAT THE ZEROAND BOUNCES OFF

RISE FALL RISE

FALL END BEHAVIOR

AGRAPHOF

n f X INCREASESWITHOUTBOUND

XDECREASES

t

arapµ

X INCREASESOF f XDECREASESWITHOUT

BOUND y

x

Example 4. The chart below illustrates the end behavior of a polynomial function:

4

f J jf i f i

fv v

gt i f i

t t tv v

Example 5. Choose the end behavior of the polynomial function:

f(x) = �(x+ 7)6(x+ 5)

4(x� 5)

3(x� 7)

3

5

NEGATIVE LEADINGCOEFFICIENT

DEGREE OF POLYNOMIAL IS THE SUMOF THE

MULTIPLICITIES 6 t 4 t 3 t 3 16

EVEN DEGREE NEGATIVE LEADING COEFFICIENT

L s

t tv

Example 6. Choose the option below that describes the behavior at x = �3 of the polynomial:

f(x) = (x+ 6)(x+ 3)4(x� 3)

3(x� 6)

6

x 6 3 3 x 6

ZEROSI X 6 ODDMULTIPLICITY

2 X 3 EVEN MULTIPLICITY

X 3 ODDMULTIPLICITY

4 X 6 ODDMULTIPLICITY

A POSITIVELEADINGCOEFFICIENTDEGREEOF f X ISTHE SUM OF THE MULTIPLICITIES

I t 4 t 3 I 9ODD DEGREE AND POSITIVE LEADING COEFFICIENT

9

L

PLOTZEROSAND SKETCHGRAPH

Example 7. Choose the option below that describes the behavior at x = �9 of the polynomial:

f(x) = (x+ 9)3(x+ 3)

5(x� 3)

3(x� 9)

2

7

4 x 3 x 3 X 4

ZEROS

i ii 03 X 3 ODDMULTIPLICITY4 X 4 EVEN MULTIPLICITY

POSITIVELEADINGCOEFFICIENTDEGREEOF f X ISTHE SUM OF THE MULTIPLICITIES

3 5 t 3 t 2 13ODD DEGREE AND POSITIVE LEADING COEFFICIENT

9

I

LOTZEROSAND SKETCHGRAPH

4.51ft

Example 8. Choose the end behavior of the polynomial function:

f(x) = (x+ 9)(x+ 4)6(x� 4)

3(x� 9)

8

0

A POSITIVELEADINGCOEFFICIENTDEGREEOF f X ISTHE SUM OF THE MULTIPLICITIESI t Gt3 t l 11

ODD DEGREE AND POSITIVE LEADING COEFFICIENT

t

lvv

6.2 Graphing Polynomials

Definition

A of the graph of a polynomial function is the point

where a function changes from rising to falling or from falling to rising. A polynomial of degree n

will have at most turning points.

How to Determine the Zeros and Multiplicities of a Polynomial of Degree n Given its

Graph

1. If the graph crosses the x-axis at the intercept, it is a zero with

.

2. If the graph touches the x-axis and bounces o↵ the axis, it is a zero with

.

3. The sum of the multiplicities is .

Definition

If a polynomial of lowest degree p has x-intercepts at x = x1, x2, ..., xn, then the polynomial can

be written in factored form:

.

9

TURNING POINT

n 1

ODD

MULTIPLICITY

EVEN MULTIPLICITY

h

f X X X X XzP X Xn Ph

9p p tpzt tpn

Note 6. In the factored form of a polynomial, the powers on each factor can be determined by the

behavior of the graph at the corresponding , and the stretch factor a can

be determined given a value of the function other than the - .

How to Determine a Polynomial Function Given its Graph

1. Identify the - to determine the factors of the polynomial.

2. Examine the behavior of the graph at x-intercepts to determine the

of each factor.

3. Find the polynomial of least degree containing all the factors found in step 2.

4. Use any other point on the graph (typically the y-intercept) to determine the stretch factor

(or, you can analyze the end behavior of the graph to determine the stretch factor).

10

ZEROSX INTERCEPTS

INTERCEPTS

MULTIPLICITY

IF x C IS A ZERO THEN X C IS AFACTOROFTHE POLYNOMIAL

f a x cmx Cdm x CmnInn

G Cz Cmn ARE THE ZEROSAND X INTERCEPTS

my Mn ARE THE MULTIPLICITIESMit Mzt tmn M ISTHE DEGREE OF f X4 IS THE LEADING COEFFICIENT

Example 9. Write an equation of the function graphed below:

11

X INTERCEPTS

1 X D CROSSESX AXIS ODDMULTIPLICITY

2 XI l CROSSES X AXIS ODDMULTIPLICITY

3 11 2 CROSSES X AXIS ODDMULTIPLICITYf X G X o X l X Z

a X X l x 2

END BEHAVIOR

DEGREE 3ODD DEGREE AND I

I f X XIX l x 2

I IPOSITIVE LEADING COEFFICIENTat

Example 10. Write an equation of the function graphed below:

12

X INTERCEPTS

1 X I CROSSES X AXIS ODDMULTIPLICITYX C l Xt 1 IS A FACTOR WITHMULTIPLICITY 1

2 X O CROSSES X AXIS ODDMULTIPLICITYX O X IS A FACTORWITH MULTIPLICITY 1

3 X 2 CROSSES X AXIS ODDMULTIPLICITYX 2 IS A FACTORWITH MULTIPLICITY 1

END BEHAVIOR

DEGREE 3 qODD DEGREE AND END BEHAVIOR IS y e

e I

NEGATIVE LEADING COEFFICIENT SO a co 4 1

f X XH X X Z

t

How to Sketch the Graph of a Polynomial Function

1. Find the x-intercepts (zeros).

2. Find the y-intercepts.

3. Check for symmetry. If the function is an even function, then its graph is symmetric about

the - (that is, f(�x) = f(x)). If the function is an odd function,

then its graph is symmetric about the - (that is, f(�x) = �f(x)).

4. Determine the behavior of the polynomial at the zeros using their .

5. Determine the .

6. Sketch a graph.

7. Check that the number of does not exceed one less

than the degree of the polynomial.

13

Y AXISX AXIS

MULTIPLICITIES

LEADING COEFFICIENT

TURNING POINTS

6.3 Lowest Degree Polynomial

The Factor Theorem

k is a zero of f(x) if and only if is a factor of f(x).

Note 7. The following statements are equivalent:

Note 8. If we are given the zeros of a polynomial, we can use the

to construct the lowest-degree polynomial.

Example 11. Construct the lowest-degree polynomial given the zeros below:

3,�3,�4

14

X K

X _9 IS A ZERO OFTHE FUNCTION f

2 11 9 IS A SOLUTION OFTHE EQUATION f1 3 0

3 X 4 IS A FACTOR OF THE POLYNOMIAL f X

4 Clio IS AN X INTERCEPTOFTHE GRAPHOF f X

FACTOR THEOREM

FACTORS

1 X 3IfCx x 3 Xt 3 Xt4 1

2 X C3 Gt3f x 42 3 3 4 4

FOIL3 X C 4 Xt 4 f x X 9 Xt4

fCx X3t4xt_9


�4

3,�3

2,�3

Fundamental Theorem of Algebra

If f(x) is a polynomial of degree n > 0, then f(x) has at least one

. In fact, if f(x) is a polynomial of degree n > 0 and a is a nonzero real number,

then f(x) has exactly n :

where c1, c2, ..., cn are complex numbers. That is, f(x) has if we allow for

multiplicities.

Note 9. This does NOT mean that every polynomial has an imaginary zero. Real numbers are a

subset of the complex numbers, but complex numbers are not a subset of the real numbers.

15

FACTORS

1 X l 43 Xt't GET RIDOF FRACTIONS Xt43 0 3 3 4 0

13

2 X f Zz X t GET RIDOF FRACTIONS X t O 2 2 3 0

3 X C3 1 13737

flX xt3 3xt4 2Xt

COMPLEX

ZERO

LINEAR FACTORS

f X G X C X Cz X Cn

h ROOTS

The Linear Factorization Theorem

If f is a polynomial function of degree n, then f has n , and each factor is of the

form , where c is a complex number. That is, a polynomial function has the same

number of linear factors as its degree.

Complex Conjugate Theorem

Suppose f is a polynomial function with real coe�cients. If f has a complex zero of the form a+bi,

then the of the zero, a� bi, is also a zero.

A Closer Look at the Zeros of a Polynomial Function

Recall the quadratic formula:

x =�b±

pb2 � 4ac

2a.

Case 1: b2 � 4ac is positive and not a perfect square:

x = ±

x = ±

Case 2: b2 � 4ac is negative:

x = ±

x = ±

16

FACTORSX c

COMPLEX CONJUGATE

peg

INTEGER IRRATIONALINTEGER INTEGER

RATIONAL IRRATIONAL

INTEGERINTEGER

COMPLEXINTEGER

RATIONAL COMPLEX


p2,

1

3


4 + 3i,�2

5

17

SINCEFL IS A ZERO JE IS ALSO A ZEROFACTORS

1 X Fr2 x tf Xtra3 X t GETRIDOFFRACTION X 5 013 3 1 0

f x X F Xtc 3 1f X 1 2 1 Fix fix 2 3 1f X 1 2 2 3 1

fCX 3x3_x2 6x

SINCE 4t3i IS A ZERO 4 3i IS ALSO A ZEROFACTORS

1 X 4 33 1 4 34 12 x 4 34 11 437 13 X l F Xt GET RID OFFRACTIONS X t o 5

122410

1f x X 4 3 i x 4 3 i 5 1 2

f X x 4 3i Ix 4 t3i 5 1 2

f X x 4 2 3 if 5 1 2

f x X 8 16 Gi2 5 2

f x H2 8 25 5 1 2

sof x 5 3 2 2 40 2 16 125

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Module 6 Lecture Notes - People