chapter 3: polynomial functions


Chapter 3: Polynomial Functions

3.1 Complex Numbers3.2 Quadratic Functions and Graphs3.3 Quadratic Equations and Inequalities3.4 Further Applications of Quadratic Functions and Models3.5 Higher Degree Polynomial Functions and Graphs3.6 Topics in the Theory of Polynomial Functions (I)3.7 Topics in the Theory of Polynomial Functions (II)3.8 Polynomial Equations and Inequalities; Further

Applications and Models


3.3 Quadratic Equations and Inequalities

• Consider the– zeros of– x-intercepts of– solution set of

• Each is solved by finding the numbers that make

Quadratic Equation in One Variable

An equation that can be written in the form

ax2 + bx + c = 0

where a, b, and c are real numbers with a 0, is a quadratic equation in standard form.

1642)( 2 xxxP1642)( 2 xxxP

.01642 2 xx

. true01642 2 xx


3.3 Zero-Product Property

• If a and b are complex numbers and ab = 0, then a = 0 or b = 0 or both equal zero.

Example Solve

Solution

.01642 2 xx

01642 2 xx0822 xx0)2)(4( xx 02or 04 xx

2 or 4 xx

The solution set of the equation is {– 4,2}.

Note that the zeros of P(x) = 2x2 + 4x – 16 are –4 and 2, and the x-intercepts are (– 4,0) and (2,0).


3.3 Example of a Quadratic with One Distinct Zero

Example Solve

Solution

Graphing Calculator Solution

.0962 xx

0962 xx

3or303or03

0)3( 2

xxxx

x

There is one distinct zero, 3. It is sometimes called a double zero, or double solution (root) of the equation.


3.3 Solving x2 = k

kxkxkxkx

or0or00))((

02

2

kxkxkx

kx

Square Root Property

The solution of is kx 2

.0 if}{(c)0 if}0{(b)0 if}{(a) kkikkk

Figure 26 pg 3-38


3.3 Examples Using the Square Root Property

Solve each equation.(a) (b)

Graphing Calculator Solution (a) (b)

102 2 x 14)1(2 2 x

552

ixx

71717)1( 2

xxx

This solution is an approximation.


3.3 The Quadratic Formula

• Complete the square on and rewrite P in the form To find the zeros of P, use the square root property to solve for x.

cbxaxxP 2)(.)()( 2 khxaxP

aacbb

aacb

abx

aacb

abx

aacb

abx

ac

ab

abx

abx

acx

abx

acx

abx

cbxax

24

24

2

44

2

44

2

44

0

0

22

2

2

2

22

2

2

2

22

2

2

2


• The expression under the radical, is called the discriminant.

• The discriminant determines whether the quadratic equation has – two real solutions if – one real solution if– no real solutions if

3.3 The Quadratic Formula and the Discriminant

The Quadratic Formula

The solutions of the equation , where , are

0a02 cbxax

.2

42

aacbb

x

,42 acb

,042 acb,042 acb.042 acb


3.3 Solving Equations with the Quadratic Formula

SolveAnalytic Solution

.22)2( xxx

22)2( xxx

024222

2

2

xxxxx Rewrite in the form

ax2 + bx + c = 0

Substitute into the quadratic formula a = 1, b = – 4, and c = 2, and simplify.

)1(2)2)(1(4)4()4( 2

x

222

2242

842

8164

x


3.3 Solving Equations with the Quadratic Formula

Graphical Solution

With the calculator, we get an approximation for .22 and 22


3.3 Solving a Quadratic Equation with No Solution

Example Solve

Analytic Solution

Graphing Calculator Solution

.042 2 xx

431

41

4311

4311

43211

)2(2)4)(2(4)1()1( 2

ii

x

Figure 30 pg 3-43


• Similar for a quadratic function P(x) with a < 0.

P(x) a>0

x-axis

Solution Set of Is P(x)=0 {a,b} P(x)<0 interval (a,b) P(x)>0 (a,0) (b,0)

3.3 Possible Orientations for Quadratic Function Graphs (a > 0)

),(),( ba

P(x) a>0

x-axis

Solution Set of Is P(x)=0 {a} P(x)<0 P(x)>0

(a,0)

P(x) a>0

x-axis

Solution Set of Is P(x)=0 P(x)<0 P(x)>0

a ),(

),(),( aa


3.3 Solving a Quadratic Inequality Analytically

Solve

The function is 0 when x = – 3 or x = 4. It will be greater than 0 or less than zero for any other value of x. So we use a sign graph to determine the sign of the product (x + 3)(x – 4).

For the interval (–3,4), one factor is positive and the other is negative, giving

a negative product. The product is positive elsewhere since (+)(+) and (–)(–) is positive.

.0122 xx

0)4)(3(0122

xxxx

4or3 xx


3.3 Solving a Quadratic Inequality Graphically

Verify the analytic solution to Y1 < 0 in the interval (–3,4).

:0122 xx

Solving a Quadratic Inequality1. Solve the corresponding quadratic equation.2. Identify the intervals determined by its solutions.3. Use a sign graph to determine the solution interval(s).4. Decide whether or not the endpoints are included.


3.3 Literal Equations Involving Quadratics

Example Solve (a) (b)

Solution(a)

(b)

dd for 4

A2 .for )0(2 trkstrt

4A

2d 24A d

A42 d

A2A4 d

kstrt 2 02 kstrt

kcsbraa

acbbt ,, where2

42

2

))((4)()( 2

rkrsst

2

42

rrkss


3.3 Modeling Length of Life

The survival rate after age 65 is approximated bywhere x is measured in decades. This function gives the probability that

an individual who reaches age 65 will live at least x decades (10x years) longer. Find the age for which the survival rate is .5 for people who reach the age of 65. Interpret the result.

Solution Let Y1 = S(x), Y2 = .5, and find the positive x-value of an intersection point.

,076.058.1)( 2xxxS

Discard x = – 3. For x 2.2, half the people who reach age 65 will live about 2.2 decades or 22 years longer to age 87.

chapter 3: polynomial functions

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