OUTLINEOUTLINE

Homework (and homework questions)Homework (and homework questions) Ask any review questions you wantAsk any review questions you want Review long division, solve by factoring Review long division, solve by factoring

and graphing calculatorsand graphing calculators BREAK BREAK 6.5 Notes (not on quiz)6.5 Notes (not on quiz) Homework time / study timeHomework time / study time

Homework:Homework:

Quiz Friday! About 40-50 Quiz Friday! About 40-50 ptspts

Key is online! Class notes are online!Key is online! Class notes are online! Questions?Questions?

Review Long divisionReview Long division)22()8642( 23 xxxx

Review Solve by Review Solve by FactoringFactoring

27125 3 x

Graphing calculatorsGraphing calculators

Find all the real zeros of each Find all the real zeros of each

How many roots or zeros should each How many roots or zeros should each have?have?

32 3 xy 13 34 xxy

Section 6.5Section 6.5

Rational Root Theorem!Rational Root Theorem!

ObjectivesObjectives

Use the rational, irrational, and imaginary Use the rational, irrational, and imaginary root theorems to locate root theorems to locate allall the roots of a the roots of a given polynomial equationgiven polynomial equation

Rational Root TheoremRational Root Theorem This only works if we have integer This only works if we have integer

coefficients.coefficients.

Possible rational roots = Possible rational roots =

q

p

Find the rational roots ofFind the rational roots of

List the List the possible rational roots:possible rational roots:

Test using substitution or synthetic Test using substitution or synthetic divisiondivision

03323 xxx

Use the rational root Use the rational root theorem to find ALL rootstheorem to find ALL roots

0122 23 xxx

Use the rational root Use the rational root theorem to find ALL rootstheorem to find ALL roots

01052 23 xxx

The Irrational Root The Irrational Root TheoremTheorem

If is a root of a polynomial If is a root of a polynomial equation with rational coefficients, then equation with rational coefficients, then the conjugate is also a rootthe conjugate is also a root

In other words, irrational roots occur in In other words, irrational roots occur in conjugate pairs. conjugate pairs.

If is a root, then is a rootIf is a root, then is a root

ba

ba

3 3

The Imaginary Root The Imaginary Root TheoremTheorem

If the imaginary number If the imaginary number aa + + bibi is a root of is a root of a polynomial equation with real a polynomial equation with real coefficients, then the conjugate coefficients, then the conjugate a – bi a – bi also is a rootalso is a root

Imaginary roots Imaginary roots alsoalso occur in conjugate occur in conjugate pairspairs

If 3, and are roots of a If 3, and are roots of a polynomial with integer coefficients, find polynomial with integer coefficients, find two additional roots.two additional roots.

What is the degree of this polynomial?What is the degree of this polynomial?

i27

Find a third-degree polynomial equation Find a third-degree polynomial equation with rational coefficients that has roots 3 with rational coefficients that has roots 3 and 1 + and 1 + ii

HomeworkHomework

Page 333 (1-3), 7, 8, 11, (16-18), 23, 24 Page 333 (1-3), 7, 8, 11, (16-18), 23, 24 (32-36) even(32-36) even

Due MondayDue Monday