Tuesday, June 2, 2009

Polynomials

NOTE: This is Chapter 2, Section 2. Reviewing Chapter 2, Section 1 before coming here might not be a bad idea. For really quick review, go to the bottom section, labeled..."Polynomial Review and Theorems."

A polynomial function of nth degree has at most n roots.

has 2 roots.

has 3 roots.

has 4 roots.

Descartes' Rule of Signs:
Let f(x) be a polynomial function with real coefficients.



The max number of positive real roots of f(x) equals the number of sign changes between consecutive coefficients or less that number by an even integer.

Ex. 4 -> 2 -> 0


The max number of negative real roots equals the number of sign changes between consecutive coefficients of f(-x) or less that number by an even integer.

Ex. 3 -> 1

A variation in sign means that consecutive (nonzero) coefficients have opposite signs.


Ex.
at most 2 positive roots. (sign change between 1x and -4x, and -4x,1x)

at most 1 negative roots. (sign change between -1x and +12)



Rational Roots of Polynomials:
Rational Zero Test:
If



The possible rational roots of f(x) must be in form:







Ex. Find the rational zeros of



Solution: The leading coefficient is 2 and the constant term is 3.

Possible rational zeros:


By synthetic division, you can determine that is a rational zero. So f(x) factors as and you can calculate the rational zeros of f to be




To Determine the Number of Zeros of a Polynomial Function:
The Fundamental Theorem of Algebra and The Linear Factorization Theorem are EXISTENCE theorems; they tell you how many zeros a polynomial has, but not HOW to find them.



The Fundamental Theorem of Algebra: If f(x) is a polynomial of degree n, where n is greater than zero, then f has at least one zero in the complex number system (that would be the realm of imaginary numbers).


Linear Factorization Theorem:

If f(x) is a polynomial of degree n where n is greater than zero, then f has
precisely n linear factors where are complex numbers. N zeros may be real or complex, and may be repeated.




Real Zeros of a Polynomial Function:
Don't forget multiplicity:
has two zeros




Real and Complex Zeros of a Polynomial Function:

has the zeros because when factored completely as ,



-->
-->

Though Descartes' Analysis will not immediately identify imaginary roots, it is very helpful in reducing the equation; imaginary roots are generally easily solved for once the real rationals are found using synthetic division (i.e., use the quadratic formula).



Finding the Zeros of a Polynomial Function
To write

as a product of linear factors and list the zeros:


The possible rational zeros are
(if you don't understand where this came from, go back and re-read "the rational zero test"). Using synthetic division, you can determine that -2 is a zero and 1 is a repeated zero. SO:


can be factored as
which finally comes out to


as the factored equation and

as the zeros.


Polynomial Review and Important Theorems:

Intermediate Value Theorem
: If 2 consecutive x values yield y values of opposite sign (+ -) or (- +), then there is a real root between the x-values.


Remainder Theorem
: When f(x) is divided by (x-k), the remainder is the value f(k)


Factor Theorem:
If (x-k) is a factor, then k is a root.



Rational Zero Test
: The possible rational roots of a polynomial function
are


Descartes’ Rule of Signs
: For a polynomial function
, the number of :
Positive roots = the number of sign changes in f(x) or less that number by an even integer.

Negative roots
= the number of sign changes in f(-x) or less that number by an even integer.

An nth degree polynomial equation has n roots.



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Practice Problems

1) Write an equation given the roots






2) Write the equation of a polynomial function whose possible rational roots are:





3) Identify the maximum number of positive real and negative real roots in the equation:





4) Write the equation for the following graph:










5) One root of is -3. Find the other roots.






6) Find the rational and irrational roots of







7) Question: Find the equation of the polynomial graph:











8) Find all roots of






9) Give the equation for this graph:





10) Given equation , find the zeros.





11) Find all roots of



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SOLUTIONS:




1)





2)





3) Max +real roots: 2
Max -real roots: 2





4)






5)





6)







7)




8)




9)




10) Zeros:




11)

4 comments:

kelseyyyy said...

How do you tell the difference between possible positive rational roots and negative?

Daniela M said...

for the remainder theorem- what is (x-k) and how do you use f(k) in the problems?

Kate Nowak said...

Kelsey - The rational roots can be either positive or negative. Sometimes you can narrow it down with the IVT. For example, if a graph contains the points (1,3) and (2, -5), it must have a root between 1 and 2.

Daniela - If k is a root of P(x), then (x-k) will divide into P(x) with no remainder.

If k is not a root, then P(x) divided by (x-k) will have a remainder of f(k).

For example: What is the remainder when f(x) = x^3 + x^2 + x + 1 is divided by (x-2)? Answer: 15, because f(2) = 15.

Kate Nowak said...

Also think about it this way, if k is a root, then P(k) = 0.