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)