Polynomial

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:

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These aren't polynomials:

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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.

Binomial Theorem

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

Intermediate and Olympiad Topics

Other Important Topics

• Vieta's formulas
• Newton's identities

Other Resources

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

See also

• Algebra