2-4 zeros of a polynomial function

2-4Zeros of a Polynomial Function

Chapter 2Power, Polynomial, and Rational Functions

Warm-up

Factor using long division.1.

Find f(c) using synthetic substitution.2.

3 ;968204 23 xxxx

4 ;382423)( 3456 cxxxxxxf

Homework Check

…and a short Homework Quiz

Complete the following:

Complex Numbers(2 – 3i, 2i, 16, )

Recall the values of the following:

i = i2 = i3 = i4 = i5 = i6 = i7 =

•Fractions•Integers•Irrational numbers•Imaginary numbers• Natural numbers•Rational numbers•Real numbers•Whole numbers

1

Complex Numbers(2 – 3i, 2i, 16, )

Real Numbers

Rational Numbers

Fractions Integers

Whole Numbers

Natural Numbers

Irrational Numbers

Imaginary

Numbers

Recall the values of the following:

i = i2 = i3 = i4 = i5 = i6 = i7 =

•Integers•Irrational numbers•Imaginary numbers• Natural numbers•Rational numbers•Real numbers•Whole numbers

1

Objectives for 2-4

Find real zeros of polynomial functions Find complex zeros of polynomial

functions

1. Real Zeros of a Polynomial The leading coefficient and constant

term with integer coefficients can be used to determine a list of all possible rational zeros.

Then you can determine actual zeros using synthetic division.

This is the Rational Zero Theorem

Rational Zero TheoremEvery rational zero of a polynomial has the

form , where p is an integer factor of the constant

term q is an integer factor of the leading

coefficient

qp

Example 1: List all possible rational zeros. Then determine which, if any, are zeros.

423)( 23 xxxxf423)( 23 xxxxf

Example 2: List all possible rational zeros. Then determine which, if any, are zeros.

12)( 3 xxxf

Example 3: List all possible rational zeros. Then determine which, if any, are zeros.

152852)( 3 xxxxf152852)( 3 xxxxf

2. Writing a Polynomial Given its Zeros

Write a polynomial function of least degree with real coefficients in standard form that has -1, 2, and 2 – i as zeros.

That’s enough for one day… Practice these skills, and then we will

put everything together after the mid-chapter quiz next class.

The Mid-chapter quiz on Tuesday…Topics covered (2 – 1 through 2 – 3): Domain and Range of graphed functions Solving radical equations Determining end behavior of a polynomial

without the use of a calculator Determining the number of turning points

and where functions increase and decrease

Using long division, synthetic division, and synthetic substitution to determine factors of polynomial functions

Assignment due Thursday Practice with skills from today’s

lessonp. 127, #3, 5, 11, 13, 15, 33, 35, 37.

Finish the Mid-chapter quiz

Data analysis

38 48 58 68 78 88 98 10850

60

70

80

90

100

f(x) = 0.391705456620319 x + 46.4127211027358R² = 0.481288947669606

Quarter 1 Average

Homework Grade

Qua

rter

1 A

vera

ge

Almost 14% of the variability in overall grade can be attributed to the homework grade.

38 48 58 68 78 88 98 10850

60

70

80

90

100

f(x) = 0.176381049945879 x + 63.9334425416216R² = 0.135776830314401

Quarter 1 Average

Homework Grade

Qua

rter

1 A

vera

ge

Almost half of the variability in overall grade can be attributed to your homework average.

38 48 58 68 78 88 98 10850

60

70

80

90

100

f(x) = 0.391705456620319 x + 46.4127211027358R² = 0.481288947669606

Quarter 1 Average

Homework Grade

Qua

rter

1 A

vera

ge

Homework Check:

A couple more concepts will be of some help in pulling all the ideas together in this section.

Upper and Lower Bounds Tests (EASY!)

Descartes’ Rule of Signs (non-essential)

Pull it all together with the first half of this lesson

Using the Upper and Lower Bounds Test To narrow the search for real zeros,

you can determine the interval in which the real zeros are located.

This function seems have real zeros between -2 and 2. Eliminate all zeros outside that interval.How easy is that!?

Lower Bound Upper Bound

Descartes’ Rule of Signs tells you the number of positive or negative real zeros

If you are interested, you can find this on p. 123 in your textbook.

It is non-essential, but an interesting theoretical construct by the great French mathematician Decartes.

Pulling it all together!

Factor and find the zeros (both real and irreducible quadratic factors)

Then, factor the irreducible quadratic factors into imaginary roots and list all the zeros.

Example:

Write k(x) as the product of linear and irreducible quadratic factors.

Write k(x) as the product of linear factors.

List all the zeros of k(x).

24142313)( 2345 xxxxxxk

Example:Write k(x) as the product of linear and irreducible quadratic factors.

Step 1: List all possible factors 24142313)( 2345 xxxxxxk

Example:Write k(x) as the product of linear and irreducible quadratic factors.

Step 2 (optional) Check Descartes Rule of Signs

24142313)( 2345 xxxxxxk

Example:Write k(x) as the product of linear and irreducible quadratic factors.

Step 3: Look at the graph in your calculator and find upper and lower bounds of the real roots.

Eliminate all possible roots outside of the upper and lower bounds.

Start testing with those.

24142313)( 2345 xxxxxxk

Example:Write k(x) as the product of linear and irreducible quadratic factors.

The graph suggests that 4 is a zero. Start there.

Use the depressed polynomial to test the next possible zero.

24142313)( 2345 xxxxxxk

Example:Write k(x) as the product of linear and irreducible quadratic factors.

The graph suggests that -2 is another zero. Try that one.

Use the depressed polynomial to test the next possible zero.

6575 234 xxxx

Example:Write k(x) as the product of linear and irreducible quadratic factors.

The graph suggests that -3 is another zero. Try that one.

Write the depressed polynomial. Note that it is irreducible (It can’t be factored with real roots).

33 23 xxx

Summarize all the roots and factors Roots: 4, -2, and -3

Factors:

Assignment: p. 127, 39 – 47 odds