2.6 Theorems About Roots of Polynomials

PG 98

Objectives

• Use the Rational Root Theorem to list all possible roots of a polynomial

• Test possible roots to find actual roots

• Use Conjugate Root Theorem to find pairs of roots

Rational Root Theorem

• For any polynomial with integer coefficients there are only a limited number of possible roots.

• Rational roots must have reduced from where is a factor of the constant (last term) and is a factor of the leading coefficient (first term).

Rational Root Theorem• PROBLEM 1

List all of the possible rational roots for each function.A)

Rational Root Theorem• PROBLEM 1

List all of the possible rational roots for each function.B)

Rational Root Theorem

Every actual rational root must come from this list ().What is a root?

How can we tell if an item from our list is a root?

Rational Root Theorem

A root of a polynomial is a value for such that .In other words, if a possible root is in fact a root, then the polynomial will equal zero when that root is plugged in.EX: POSSIBLE ROOTS: TEST Not a RootTEST It is a root

• If a problem asks for “all possible rational roots” then you list all the p/q’s• If a problem asks for “rational roots” then:

1. Use your graphing calculator to find 1 root (x= some number like 3).2. Use that root for synthetic division (plug 3 into the box) to break down the

function.3. Repeat steps 1 and 2 if still have a cubic or higher degree polynomial, until

get down to a quadratic polynomial.4. Now factor or use the quadratic formula to break down to find all the

factors and remaining roots.

Rational Root Theorem• PROBLEM 1 (Teacher Example)

List all of the actual rational roots for each function.C)

• p. 99 Got It? and Practice 1,2

Rational Root Theorem

• PROBLEM 2: Got itList all of the actual rational roots for each function.D)

Conjugate Root Theorem

• If a polynomial has only rational coefficients then any irrational roots occur in conjugate pairs. In addition any complex roots occur in conjugate pairs.• In other words, radical and imaginary roots always show up in pairs.

• EX:

Conjugate Root Theorem

• PROBLEM 3 (teacher example)• A polynomial of degree 5 has rational coefficients. If and are all roots

of what are the other roots?

Conjugate Root Theorem• p. 100 Got It? and Practice 5,6

Problem 4 Got it: Pg 101Replace with the below example

HW 2.6Pg 103-105# 11-15, 17a, 17b, 19, 20, 24, 25