Polynomial

Definition

A polynomial is an algebraic expression built from variables, coefficients, addition, subtraction, multiplication, and nonnegative integer powers.

Example:

3x^2 + 2x + 1

Polynomial = sum of terms with nonnegative whole-number powers.

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Terms

A polynomial is made of terms.

Example:

3x^2 + 2x + 1

Terms:

3x^2
2x
1

Each term is a piece being added or subtracted.

Terms are the building blocks of a polynomial.

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Coefficients

A coefficient is the number multiplying a variable part.

Example:

3x^2

The coefficient is:

3

In:

2x

the coefficient is:

2

Coefficients control the size of terms.

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Powers

Polynomials use nonnegative integer powers.

Allowed:

x^0
x^1
x^2
x^3

Not polynomial powers:

x^-1
x^(1/2)

Those introduce different kinds of expressions.

Polynomial powers must be whole-number powers greater than or equal to zero.

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Simple Examples

Polynomials:

x + 1
x^2 + 3x + 2
5
4x^3 - x

Not polynomials:

1 / x
sqrt(x)
2^x

Polynomials have a restricted but useful structure.

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Final Summary

• A polynomial is an algebraic expression made from terms.
• Terms can contain variables raised to nonnegative integer powers.
• Coefficients multiply variable parts.
• Constants can be polynomial terms.
• 1 / x, sqrt(x), and 2^x are not polynomials.