chapter 11 polynomial functions 11.1 polynomials and polynomial functions

Chapter 11Polynomial Functions

11.1 Polynomials and Polynomial Functions

Chapter 11Polynomial Functions

11.1

Polynomials and Polynomial Functions

A polynomial function is a function of the form

f (x) = an x n + an – 1 x

n – 1 +· · ·+ a 1 x + a 0

Where an 0 and the exponents are all whole numbers.

A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

For this polynomial function, an is the leading coefficient,

a 0 is the constant term, and n is the degree.

an 0

an

an leading coefficient

a 0

a0 constant term n

n

degree

descending order of exponents from left to right.

n n – 1

Objective: Determine whether a number is a root or zero of a given equation or function.

Objective: Determine whether a number is a root or zero of a given equation or function.

Objective: Determine whether a number is a root or zero of a given equation or function.

Objective: Determine whether one polynomial is a factor of another by division.

Objective: Determine whether one polynomial is a factor of another by division.

Objective: Determine whether one polynomial is a factor of another by division.

HW #11.1Pg 483-484 1-21 Odd, 22-31, 35-36

Chapter 11 Polynomial Functions

11.2 Factor and Remainder Theorems

P(10) is the remainder when P(x) is divided by x - 10.

P(10) = 73,120 P(-8) = -37, 292

Find P( -4)

Yes No Yes

We look for linear factors of the form x - r. Let us try x - 1.

We know that x - 1 is not a factor of P(x). We try x + 1.

To solve the equation P(x) = 0, we use the principle of zero products.

P(x) = (x – 2)(x + 3)(x + 5) x = 2 x = -3 x = -5

f ( x ) D( x )Q( x ) R f x x Q x( ) ( 1) ( ) 0

f Q( 1) ( 1 1) ( 1) 0 a7( 1) ( 1) 2 0

a 3 f x x x 7( ) 3 2

f 7(2) 2 3(2) 2 120

3 23 5 2 1x x bx ( x )Q( x )

b Q 3 22 3(2 ) (2) 5 (2 2) (2) 1

b 12

4. Solve

-5 < x< 1 or 2 < x < 3

HW #11.2Pg 488-489 1-15 Odd, 16-31

Chapter 11

11.3 Theorems about Roots

Carl Friedrich Gauss was one of the great mathematicians of all time. He contributed to many branches of mathematics and science, including non-Euclidean geometry and curvature of surfaces (later used in Einstein's theory of relativity). In 1798, at the age of 20, Gauss proved the fundamental theorem of algebra.

If a factor (x - r) occurs k times, we say that r is a root of multiplicity k

Where in the ____ did that come from?

The polynomial has 5 linear factors and 5 roots. The root 2 occurs 3 times, however, so we say that the root 2 has a multiplicity of 3.

-7 Multiplicity 2

3 Multiplicity 1

4 Multiplicity 2

3 Multiplicity 2

1 Multiplicity 1

-1 Multiplicity 1

Degree 3 3 roots 3 4x i 9x 3 4x i

Complex Roots Occur in Conjugate Pairs

Irrational Roots also come in Conjugate Pairs

Degree 6 6 roots 2 5x i

x i

1 3x

2 5x i

x i

1 3x

7 2 3 7 5i and

Degree 4 4 roots 2i -2i

1. Divide p(x) by a known root to reduce it to a polynomial of lesser degree

2. Divide the result by a different known root to reduce the degree again

3. Repeat Steps 1 and 2 until you have reduced it to degree 2, then factor or use the quadratic formula to find the remaining roots

Roots are 2i, -2i, 2, and 3.

, , 2, 1i i

2 ( 2)x x is a factor 1 ( 1)x x is a factor 3 ( 3 )x i x i is a factor The number an can be any

nonzero number.

Let an = 1.

We proceed as in Example 6, letting an = 1 Degree 5 5 roots

0x x is a factor 1 ( 1)x x is a factor

4 ( 4)x x is a factor

Multiplicity 3 means it is a factor 3 times

3 2) ( ) 6 3 10f p x x x x

5 4 3 2) ( ) 6 12 8g p x x x x x

1 2 1 2x x is a root 1 3 1 3x i x i is a root

4 3 2) ( ) 6 11 10 2h p x x x x x

3 2) ( ) 2 4 8i p x x x x

HW #11.3Pg 494-495 1-49 Odd, 59

4 3

No

No

2 3 4 3 4 ( ) ( ( ))( ( ))p x x x i x i2 1 2 2 2 ( ) ( )( )( ( ))( ( ))p x x x x i x i

1 2 1 2, , ,i i

Chapter 11

11.4 Rational Roots

List the possible rational zeros.

: 1, 2, 3, 4, 6, 12p : 1q

pq

pq

Test these zeros using synthetic division.

The roots of ƒ are -1, 3, and -4.

List the possible rational zeros.

:p :q

1 1 2 2 3 3 6 6: , , , , , , , ,

1 3 1 3 1 3 1 3pq

1 2: 1, , 2, , 3, 6

3 3pq

Test these zeros using synthetic division.

1 2: 1, , 2, , 3, 6

3 3pq

Test these zeros using synthetic division.

The roots of ƒ are -2, , and .

13

3

x = 1

x = -1

HW # 11.4Pg 499-500

1-11Odd, 13-21, 23-27 Odd

Chapter 11

11-5 Descartes’ Rule of Signs

Theorem 11-8 Descartes’ Rule Of Signs Part #1

The number of positive real zeros of a polynomial P(x)

with real coefficients isa. the same as the number of variations of the sign

of P(x), orb. Less than the number of variations of sign of P(x)

by a positive even integer

23 234 xxxxxfstarts Pos. changes Neg. changes Pos.

1 2

There are 2 sign changes so this means there could be 2 or 0 positive real zeros to the polynomial.

Determine the number of positive real zeros of the function

EXAMPLES

15 2( ) 2 5 3 6p x x x x

+ - + +

2 Sign Changes 2 or 0 Positive Real Roots

24 3 2( ) 5 3 7 12 4p x x x x x

+ - + -

4 Sign Changes 4, 2, or 0 Positive Real Roots

+

Determine the number of positive real zeros of the function

EXAMPLES

35( ) 6 2 5p x x x

+ - -

1 Sign Changes Exactly 1 Positive Real Roots

Try This Determine the number of positive real zeros of the function.

3) ( ) 5 4 5a p x x x

6 4 3 2) ( ) 6 5 3 7 2b p x x x x x x

2) ( ) 3 2 4c p x x x

Theorem 11-8 Descartes’ Rule Of Signs Part #2

The number of negative real zeros of a polynomial P(x)

with real coefficients isa. the same as the number of variations of the sign of

P(-x), orb. Less than the number of variations of sign of P(-x)

by a positive even integer

There are 2 sign changes so this means there could be 2 or 0 negative real zeros to the polynomial.

23 234 xxxxxf

starts Pos. changes Neg. changes Pos.1 2

Determine the number of negative real zeros of the function

EXAMPLES

44 3 2( ) 5 3 7 12 4p x x x x x

+ - +

4 Sign Changes 4, 2, or 0 Negative Real Roots

4 3 2( ) 5( ) 3( ) 7( ) 12( ) 4p x x x x x

4 3 2( ) 5 3 7 12 4p x x x x x

- +

Try This Determine the number of negative real zeros of the function.

3) ( ) 5 4 5 d p x x x

6 4 3 2) ( ) 6 5 3 7 2 e p x x x x x x

2) ( ) 3 2 4 f p x x x

68 67 69

If a sixth-degree polynomial with real coefficients has exactly five distinct real roots, what can be said of one of its roots?

Is it possible for a cubic function to have more than three real zeros?

Is it possible for a cubic function with real coefficients to have no real zeros?

HW #11.5Pg 503 1-32

Chapter 11

11-6 Graphs of Polynomial Functions

3.

4.

5.

First, plot the x-intercepts.Second, use a sign chart to determine when f(x) > 0 and f(x) < 0

-1 3

0

0

+

+ +

+ + +

f(0) =3, Sketch a smooth curve

+

+

+

First, plot the x-intercepts.Second, use a sign chart to determine when f(x) > 0 and f(x) < 0

-2 1

0

0

+

-

+

+

+

+

f(0) =2, Sketch a Smooth Curve

+

- +

First, plot the x-intercepts.

Second, use a sign chart to determine when f(x) > 0 and f(x) < 0

-2 -1

0

0+

--

+

-+

f(0) =-12, Sketch a Smooth Curve

+

--

3

(0, -12)

0

+

+

+

+ + - +

A

B

3 x-intercepts 3 real roots.

1 x-intercept, 1 real root 2 x-intercepts, 2 real roots.

The left and right ends of a graph of an odd-degree function go in opposite directions.

4 x-intercepts 4 real roots.

1 x-intercept, 1 real root

2 x-intercepts, 2 real roots.

The left and right ends of a graph of an even-degree function go in the same directions.

3 x-intercepts, 3 real roots.

Even Multiplicity

Odd Multiplicity

3. Factor and make a sign chart.

5. Plot this information and consider the sign chart.

HW #11.6Pg 507-508 1-22

Test Review

12

4. Solve

-5 < x< 1 or 2 < x < 3

The coefficient of xn-1 is the negative of the sum of the zeros.

HW #R-11aPg 511-512 1-22

• Prove the Remainder Theorem• Pg 489 #31• Pg 489 #32• Pg 503 #28• Find all the roots of a polynomial and use

them to sketch the graph• Find roots on your calculator• 2 parts

– No Calculator– Calculator

• 1 Day Test

The graph of 43 2 12P x x x

can cross the x-axis in no more than r points. What is the value of r?

2 7 ( )p x x

7Use the rational root theorem to prove that the

is irrational by considering the polynomial

2 4x kx 1x 1x

For what value of k will the remainder be the same when

is divided by or

The equation 2 2 0x ax b has a root of multiplicity 2. Find it.

HW #R-11bPg 513 1-16