Warm-Up: November 16, 2015

253125 342 xxxx

Homework Questions?

Zeros of Polynomial Functions

Section 2.5

Roots, Zeros, and -intercepts Functions have zeros Equations have roots Graphs have -intercepts

The following are always the same numbers: The zeros of The roots of The -intercepts of the graph of

Rational Zero Theorem If is a polynomial with integer coefficients,

then every possible rational zero is given by:

Possible rational zeros =

Factors of the constant term Factors of the leading

coefficient

Example 1 (like HW #1-8) List all possible rational zeros for

8526 234 xxxxxf

You-Try #1 (like HW #1-8) List all possible rational zeros for

169734 234 xxxxxf

Finding Zeros of Polynomial Functions1. Use the rational zero theorem to find all the

possible rational zeros.2. Use a guess-and-check method and

synthetic division to find a zero3. Use the successful synthetic division to get a

lower-degree polynomial that can then be solved to find the remainder of the zeros.

Example 3 (like HW #9-22) Find all rational zeros of 41583 23 xxxxf

You-Try #3 (like HW #9-22) Find all rational zeros of 252 23 xxxxf

Example 4 (like HW #9-22) Solve 015162 24 xxx

You-Try #4 (like HW #9-22) Solve 04652 23 xxx

Assignment Read Section 2.5 Page 302 #1-13 odd

An updated 2.5 guided notes has been posted, with a new page 3. It is now 5 pages instead of 4.

You have the rest of class to work on this assignment, or Chapter 1 Test corrections 2.1-2.4 Review

First Four Problems In Exercises 1-8, use the Rational Zero

Theorem to list all possible rational zeros for each given function.

121277)7

6254)5

619113)3

44)1

2345

234

234

23

xxxxxxf

xxxxxf

xxxxxf

xxxxf

Warm-Up: November 17, 2015 Without using your calculator, find all zeros of

12112 23 xxxxf

Homework Questions?

Linear Factorization Theorem An th-degree polynomial has complex roots

Each root can be written as a factor, An th-degree polynomial can be expressed as

the product of a nonzero constant and linear factors: 01

11 ... axaxaxaxf n

nn

n

nn cxcxcxaxf 21

Example 5 (like HW #23-28) Factor the polynomiala) as the product of factors that are irreducible

over the rational numbersb) as the product of factors that are irreducible

over the real numbersc) in completely factored form, including

complex (imaginary) numbers

229 24 xxxf

Example 5 (like HW #23-28) Factor the polynomial

229 24 xxxf

You-Try #5 (like HW #23-28) Factor the polynomiala) as the product of factors that are irreducible

over the rational numbersb) as the product of factors that are irreducible

over the real numbersc) in completely factored form, including

complex (imaginary) numbers

276 24 xxxf

You-Try #5 (like HW #23-28) Factor the polynomial

276 24 xxxf

Assignments Read Section 2.5 Page 302 #1-13 odd, Page 302 #23-27 odd Page 302 #29-35 odd, 43-55 odd (learning

next class) Use your graphing calculator to help find possible

roots for 43-55 odd

You have the rest of class to work on any of the above assignments, Chapter 1 Test corrections, or 2.1-2.4 review.

Warm-Up: November 18/19, 2015 Find all zeros of 152 24 xxxf

Homework Questions?

Finding a Polynomial When Given Zeros1. Write the basic form of a factored

polynomial:

2. Fill in each “” with a zero3. Fractions can be written with the

denominator in front of the “”4. If a complex (imaginary) number is a

zero, so is its complex conjugate5. Multiply the factors together6. Use the given point to find the value of

nn cxcxcxaxf 21

Example 6 (like HW #29-36) Find an degree polynomial function with real

coefficients with the following conditions:

You-Try #6 (like HW #29-36) Find an degree polynomial function with real

coefficients with the following conditions:

Example 7 (like HW #43-56) Find all roots of

024109941124 2345 xxxxx

You-Try #7 (like HW #43-56) Find all zeros of

10611067016 2345 xxxxxxf

Roots, Zeros, and -intercepts Functions have zeros Equations have roots Graphs have -intercepts

The following are always the same numbers: The zeros of The roots of The -intercepts of the graph of

Assignment Read Section 2.5 Page 302 #1-13 odd, Page 302 #23-27 odd Page 302 #29-35 odd, 43-55 odd

Use your graphing calculator to help find possible roots for 43-55 odd