Difference between revisions of "Polynomial"

Revision as of 04:01, 23 June 2006

A polynomail 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:

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

These aren't polynomails:

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

Introductory Topics

A More Precise Definition

A polynomail 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 Polynomails

What is a root?

A root is a value for a variable that will make the polynomail equal zero. For an example, 2 is a root of $x^2 - 4$ because $2^2 - 4 = 0$ . For some polynomails, you can easily set the polynomail 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 polynomail 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 polynomail in this form, because the roots will be $x_1,x_2,...,x_n$ . This also tells us that a polynomail can have up to $n$ distinct roots, where $n$ is its degree.

Factoring

Different methods of factoring can help find roots of polynomails. Consider this polynomail:

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

This polynomail easily factors to:

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

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

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

The Rational Root Theorem

Descartes' Law of Signs

Binomial Theorem

Binomail theorem can be very useful for factoring and expanding polynomails.

Intermediate Topics

Multiplying and Dividing Polynomials

Synthetic Division

Intermediate and Olympiad Topics

Transforming Polynomails

Other Resources

An extensive coverage of this topic is given in A Few Elementary Properties of Polynomials by Adeel Khan.

@@ Line 1: / Line 1: @@
+A polynomail 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:
+* <math>4x^2 + 6x - 9</math>
+* <math>x^3 + 3x^2y + 3xy^2 + y^3</math>
+* <math>5x^4 - 2x^2 + 9</math>
+These '''aren't''' polynomails:
+* <math>\sin^2{x} + 5</math>
+* <math>(4x+3)/(2x-9)</math>
+==Introductory Topics==
+===A More Precise Definition===
+A polynomail in one variable, is a function <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math>.  Here, <math>a_n</math> is the <math>n</math>th coefficient, and <math>n</math> is an integer.
+===Finding Roots of Polynomails===
+====What is a root?====
+A root is a value for a variable that will make the polynomail equal zero.  For an example, 2 is a root of <math>x^2 - 4</math> because <math>2^2 - 4 = 0</math>.  For some polynomails, you can easily set the polynomail 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 polynomail can be written as:
+<math>P(x) = k(x-x_1)(x-x_2)\cdots(x-x_n)</math> where <math>k</math> is a constant, and<math>n</math> is the highest power of <math>x</math> that <math>P(x)</math> contains (also called the ''degree'').  It's very easy to find the roots of a polynomail in this form, because the roots will be <math>x_1,x_2,...,x_n</math>.  This also tells us that a polynomail can have up to <math>n</math> distinct roots, where <math>n</math> is its degree.
+====Factoring====
+Different methods of [[factoring]] can help find roots of polynomails.  Consider this polynomail:
+<math>x^3 + 3x^2 - 4x - 12 = 0</math>
+This polynomail easily factors to:
+<math>(x+3)(x^2-4) = 0</math>
+<math>(x+3)(x-2)(x+2) = 0</math>
+Now, the roots of the polynomail are clearly -3, -2, and 2.
+====The Rational Root Theorem====
+====Descartes' Law of Signs====
+====Binomial Theorem====
+[[Binomail theorem]] can be very useful for factoring and expanding polynomails.
+==Intermediate Topics==
+===Multiplying and Dividing Polynomials===
+===Synthetic Division===
+==Intermediate and Olympiad Topics==
+===Transforming Polynomails===
+===Other Important Topics===
+* [[Vieta's formulas]]
+* [[Newton's identities]]
+==Other Resources==
 An extensive coverage of this topic is given in [http://www.artofproblemsolving.com/Resources/Papers/PolynomialsAK.pdf A Few Elementary Properties of Polynomials] by Adeel Khan.

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