Difference between revisions of "Polynomial"
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− | A | + | 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: | For example, these are polynomials: | ||
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* <math>5x^4 - 2x^2 + 9</math> | * <math>5x^4 - 2x^2 + 9</math> | ||
− | These '''aren't''' | + | These '''aren't''' polynomials: |
* <math>\sin^2{x} + 5</math> | * <math>\sin^2{x} + 5</math> | ||
* <math>(4x+3)/(2x-9)</math> | * <math>(4x+3)/(2x-9)</math> | ||
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===A More Precise Definition=== | ===A More Precise Definition=== | ||
− | A | + | 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 | + | ===Finding Roots of Polynomials=== |
====What is a root?==== | ====What is a root?==== | ||
− | A root is a value for a variable that will make the | + | 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==== | ||
− | The fundamental theorem of algebra states that any | + | 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 | + | <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==== | ====Factoring==== | ||
− | Different methods of [[factoring]] can help find roots of | + | Different methods of [[factoring]] can help find roots of polynomials. Consider this polynomial: |
<math>x^3 + 3x^2 - 4x - 12 = 0</math> | <math>x^3 + 3x^2 - 4x - 12 = 0</math> | ||
− | This | + | This polynomial easily factors to: |
<math>(x+3)(x^2-4) = 0</math> | <math>(x+3)(x^2-4) = 0</math> | ||
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<math>(x+3)(x-2)(x+2) = 0</math> | <math>(x+3)(x-2)(x+2) = 0</math> | ||
− | Now, the roots of the | + | Now, the roots of the polynomial are clearly -3, -2, and 2. |
====The Rational Root Theorem==== | ====The Rational Root Theorem==== | ||
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====Binomial Theorem==== | ====Binomial Theorem==== | ||
− | [[Binomial theorem]] can be very useful for factoring and expanding | + | [[Binomial theorem]] can be very useful for factoring and expanding polynomials. |
==Intermediate Topics== | ==Intermediate Topics== | ||
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==Intermediate and Olympiad Topics== | ==Intermediate and Olympiad Topics== | ||
− | ===Transforming | + | ===Transforming Polynomials=== |
===Other Important Topics=== | ===Other Important Topics=== |
Revision as of 10:15, 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:
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
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 Important Topics
Other Resources
An extensive coverage of this topic is given in A Few Elementary Properties of Polynomials by Adeel Khan.