Warm-Up: January 9, 2012. Homework Questions? Zeros of Polynomial Functions Section 2.5.

Warm-Up: January 9, 2012

253125 342 xxxx

Homework Questions?

Zeros of Polynomial Functions

Section 2.5

Rational Zero Theorem If f(x) 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

Warm-Up: January 10, 2012 Find all rational zeros of

12112 23 xxxxf

Homework Questions?

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

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

Linear Factorization Theorem An nth-degree polynomial has n complex roots

c1, c2, …, cn

Each root can be written as a factor, (x-ci) An nth-degree polynomial can be expressed as

the product of a nonzero constant and n 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

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

Warm-Up: January 11, 2012 Find all zeros of

Hint: Start by factoring as we did in Example 5.

152 24 xxxf

Homework Questions?

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

polynomial:

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

denominator in front of the “x”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 an

nn cxcxcxaxf 21

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

real coefficients with the following conditions: n = 4 Zeros = {-2, -1/2, i} f(1) = 18

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

real coefficients with the following conditions: n = 3 Zeros = 4, 2i f(-1) = -50

Warm-Up: January 12, 2012 Simplify and write in standard form (refer to

2.1 notes if needed)

i

i

35

32

Homework Questions?

Descarte’s Rule of Signs Let f(x) be a polynomial with real coefficients1. The number of positive real zeros of f is

either equal to the number of sign changes of f(x) or is less than that number by an even integer. If there is only one variation in sign, there is exactly one positive real zero.

2. The number of negative real zeros of f is either equal to the number of sign changes of f(-x) or is less than that number by an even integer. If f(-x) has only one variation in sign, there is exactly one negative real zero.

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

024109941124 2345 xxxxx

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

102112 23 xxxxf

Assignment Complete one of the following assignments: Page 302 #1-33 Every Other Odd, 43 OR Page 302 #43-55 Odd

Chapter 2 Test next week You may use a 3”x5” index card (both sides)