What is a polynomial?

A polynomial is a chain of algebraic terms with various values of powers. There are some words and phrases to look out for when you're dealing with polynomials:

Polynomial long division labelled with degree, index, co-efficient, variable and constant

  • variable - The part of an expression that can have a changing value. In 6{x^5} - 3{x^2} + 7 the variable is x
  • co-efficient - The number before a variable showing how much it is multiplied by. In 6{x^5} - 3{x^2} + 7, 6 and -3 are co-efficients
  • index - The power of a variable. In 6{x^5} - 3{x^2} + 7, the 5 of 6{x^5} and the 2 of 3{x^2} are indices (plural of index)
  • degree - The index of the highest power. In 6{x^5} - 3{x^2} + 7 the degree is 5
  • constant - A number that does not contain the variable. In 6{x^5} - 3{x^2} + 7, the constant is 7

6{x^5} - 3{x^2} + 7 is a polynomial in x of degree 5.

4{x^2} - 8x is a polynomial in x of degree 2.

Is it a polynomial?

For the expression to be a polynomial the index of any variable must be a positive whole number

Question

Is 8\sqrt x  + 7{x^2} a polynomial?

No. 8\sqrt x  + 7{x^2} is not a polynomial.

Another way of writing \sqrt x is as {x^{\frac{1}{2}}}.

This index is a fraction not a whole number. That is why this is not a polynomial.

Question

Is 7{x^2} + \frac{4}{{{x^2}}} a polynomial?

No. 7{x^2} + \frac{4}{{{x^2}}} is not a polynomial.

Another way of writing \frac{4}{{{x^2}}} is 4{x^{-2}}.

This index is negative not a positive whole number, so this is not a polynomial.

curriculum-key-fact
  • For an expression to be a polynomial every index must be a positive whole number.
  • There cannot be any roots of a variable (this would be a fractional index)
  • There cannot be any variable in the denominator of a fraction (this would be a negative index)