Difference between revisions of "Polynomial"

Revision as of 11:15, 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:

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

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

Descartes' Law of Signs

Binomial Theorem

Binomial theorem can be very useful for factoring and expanding polynomials.

Intermediate Topics

Multiplying and Dividing Polynomials

Synthetic Division

Intermediate and Olympiad Topics

Transforming Polynomials

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.
+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:
@@ Line 6: / Line 6: @@
 * <math>5x^4 - 2x^2 + 9</math>
-These '''aren't''' polynomails:
+These '''aren't''' polynomials:
 * <math>\sin^2{x} + 5</math>
 * <math>(4x+3)/(2x-9)</math>
@@ Line 15: / Line 15: @@
 ===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.
+A polynomial 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===
+===Finding Roots of Polynomials===
 ====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.
+A root is a value for a variable that will make the polynomial equal zero.  For an example, 2 is a root of <math>x^2 - 4</math> because <math>2^2 - 4 = 0</math>.  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 polynomail can be written as:
+The fundamental theorem of algebra states that any polynomial 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.
+<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 polynomial in this form, because the roots will be <math>x_1,x_2,...,x_n</math>.  This also tells us that a polynomial 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:
+Different methods of [[factoring]] can help find roots of polynomials.  Consider this polynomial:
 <math>x^3 + 3x^2 - 4x - 12 = 0</math>
-This polynomail easily factors to:
+This polynomial easily factors to:
 <math>(x+3)(x^2-4) = 0</math>
@@ Line 41: / Line 41: @@
 <math>(x+3)(x-2)(x+2) = 0</math>
-Now, the roots of the polynomail are clearly -3, -2, and 2.
+Now, the roots of the polynomial are clearly -3, -2, and 2.
 ====The Rational Root Theorem====
@@ Line 49: / Line 49: @@
 ====Binomial Theorem====
-[[Binomial theorem]] can be very useful for factoring and expanding polynomails.
+[[Binomial theorem]] can be very useful for factoring and expanding polynomials.
 ==Intermediate Topics==
@@ Line 59: / Line 59: @@
 ==Intermediate and Olympiad Topics==
-===Transforming Polynomails===
+===Transforming Polynomials===
 ===Other Important Topics===

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