# Is f(x)=(2x^3-5x^5-2)/(9x^2+9) a polynomials?

Jun 29, 2015

No, the ratio is not a polynomial. (The numerator and denominator separately are polynomials.)

#### Explanation:

A polynomial (in one variable) consists of terms (things added together), each of which is a constant (a number) times the variable raised to some positive whole number power (or just a constant alone).

$2 {x}^{3} - 5 {x}^{5} - 2$ is an example of a polynomial.
The terms are $2 {x}^{3}$ and $- 5 {x}^{5}$ and $- 2$.
Actually, I've listed the terms with non0zero coefficient. If we want, we can also insert "terms" $0 {x}^{4}$ and $0 {x}^{2}$ and so on. (And even $0 {x}^{7}$ if there is a reason to add that term.)

The expression $\frac{2 {x}^{3} - 5 {x}^{5} - 2}{9 {x}^{2} + 9}$ is called a Rational expression.

(And the function $f \left(x\right) = \frac{2 {x}^{3} - 5 {x}^{5} - 2}{9 {x}^{2} + 9}$ is called a Rational Function.)