A polynomial is an algebraic expression involving many terms and can be factorised using long division or synthetic division.

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:

**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 \(2\).

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.

- 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)