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Difference between revisions of "Polynomial"

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A polynomial is a function in one or more variables that consists of a sum of variables raised to powers and multiplied by coefficients.
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A polynomial is a function in one or more variables that consists of a sum of variables raised to [[integer|integral]] powers and multiplied by coefficients.
  
 
For example, these are polynomials:
 
For example, these are polynomials:

Revision as of 12:15, 23 June 2006

A polynomial is a function in one or more variables that consists of a sum of variables raised to integral powers and multiplied by coefficients.

For example, these are polynomials:

  • $4x^2 + 6x - 9$
  • $x^3 + 3x^2y + 3xy^2 + y^3$
  • $5x^4 - 2x^2 + 9$

These aren't polynomials:

  • $\sin^2{x} + 5$
  • $(4x+3)/(2x-9)$


Introductory Topics

A More Precise Definition

A polynomial in one variable is a function $P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$. Here, $a_n$ is the $n$th coefficient, and $n$ 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 $x^2 - 4$ because $2^2 - 4 = 0$. 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

$P(x) = k(x-x_1)(x-x_2)\cdots(x-x_n)$ where $k$ is a constant, and $n$ is the highest power of $x$ that $P(x)$ contains (also called the degree). It's very easy to find the roots of a polynomial in this form because the roots will be $x_1,x_2,...,x_n$. This also tells us that a polynomial can have up to $n$ distinct roots, where $n$ is its degree.

Factoring

Different methods of factoring can help find roots of polynomials. Consider this polynomial:

$x^3 + 3x^2 - 4x - 12 = 0$

This polynomial easily factors to:

$(x+3)(x^2-4) = 0$

$(x+3)(x-2)(x+2) = 0$

Now, the roots of the polynomial are clearly -3, -2, and 2.

The Rational Root Theorem

For a polynomial $P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$, let p be a factor of $a_0$ and q a factor of $a_n$. It can be shown that if a rational root of P(x) exists, it must be of the form $\frac{p}{q}$.

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.


See also

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