# 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.

• 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)