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:
- 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.
