Romana Newton

2022-02-16

Does it follow that $r\left(x\right)+p\left(x\right)$ is a rational function (and not a polynomial)?
Suppose $r\left(x\right)\ne 0$ is a rational function (and not a polynomial)
$p\left(x\right)$ is a polynomial function of $x\in \mathbb{R}$.

Alexandra Haynes

Yes. Suppose that $r\left(x\right)+p\left(x\right)=s\left(x\right)$ were a polynomial. Then also $r\left(x\right)=s\left(x\right)=p\left(x\right)$ would be a polynomial, hence $r\left(x\right)$ is a polynomial, contradiction.