Difference between revisions of "Polynomial"
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====The Rational Root Theorem==== | ====The Rational Root Theorem==== | ||
− | + | For a polynomial <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math>, let p be a factor of <math>a_0</math> and q a factor of <math>a_n</math>. It can be shown that if a rational root of P(x) exists, it must be of the form <math>\frac{p}{q}</math>. | |
====Descartes' Law of Signs==== | ====Descartes' Law of Signs==== | ||
+ | By the Fundamental Theorem of Algebra, the maximum number of distinct factors (not all necessarily real) of a polynomial of degree n is n. This tells us nothing about whether or not these roots are positive or negative. Decartes' Rule of Signs says that for a polynomial P(x) the number of positive roots to the equation is equal to the number of sign changes in the coefficients of the polynomial, or is less than that number by a multiple of 2. The number of negative roots to the equation is the number of sign changes in the coefficients of P(-x) or is less than that by a multiple of 2. | ||
====Binomial Theorem==== | ====Binomial Theorem==== |
Revision as of 11:36, 23 June 2006
A polynomial is a function in one or more variables that consists of a sum of variables raised to powers and multiplied by coefficients.
For example, these are polynomials:
These aren't polynomials:
Contents
[hide]Introductory Topics
A More Precise Definition
A polynomial in one variable, is a function . Here, is the th coefficient, and is an integer.
Finding Roots of Polynomials
What is a root?
A root is a value for a variable that will make the polynomial equal zero. For an example, 2 is a root of because . For some polynomials, you can easily set the polynomial equal to zero and solve the equations to find roots, but in some cases it is much more complicated.
The Fundamental Theorem of Algebra
The fundamental theorem of algebra states that any polynomial can be written as:
where is a constant, and is the highest power of that contains (also called the degree). It's very easy to find the roots of a polynomial in this form, because the roots will be . This also tells us that a polynomial can have up to distinct roots, where is its degree.
Factoring
Different methods of factoring can help find roots of polynomials. Consider this polynomial:
This polynomial easily factors to:
Now, the roots of the polynomial are clearly -3, -2, and 2.
The Rational Root Theorem
For a polynomial , let p be a factor of and q a factor of . It can be shown that if a rational root of P(x) exists, it must be of the form .
Descartes' Law of Signs
By the Fundamental Theorem of Algebra, the maximum number of distinct factors (not all necessarily real) of a polynomial of degree n is n. This tells us nothing about whether or not these roots are positive or negative. Decartes' Rule of Signs says that for a polynomial P(x) the number of positive roots to the equation is equal to the number of sign changes in the coefficients of the polynomial, or is less than that number by a multiple of 2. The number of negative roots to the equation is the number of sign changes in the coefficients of P(-x) or is less than that by a multiple of 2.
Binomial Theorem
Binomial theorem can be very useful for factoring and expanding polynomials.
Introductory Topics
Multiplying and Dividing Polynomials
Synthetic Division
Intermediate and Olympiad Topics
Transforming Polynomials
Other Important Topics
Other Resources
An extensive coverage of this topic is given in A Few Elementary Properties of Polynomials by Adeel Khan.