Arithmetic with Rational Expressions

In What Follows…

We will be discussing how to perform arithmetic operations on what are called rational expressions.

Each of the following are examples of rational expressions:

$$\begin{align}\frac{x^2+2x+5}{x+1}&, & \frac{-5x^4-3x+6}{3.14x^2+5.6x-3} &, &\frac{x-1}{\frac{1}{2}x^2-5x+\frac{2}{3}}\end{align}$$

It is also worth noting that any polynomial is also technically a rational expression, because you can always write a $1$ in the denominator, and constants (numbers) are polynomials. For example,

$$\begin{align}5x^2+3x-2&=\frac{5x^2+3x-2}{1}\end{align}$$

Throughout the following topics, it helps to keep in mind that, since variables like $x$ represent unknown numbers, polynomials are therefore also unknown numbers (for instance, if I knew what $x$ was in the expression above, then I would be able to figure out the value of $5x^2+3x-2$). Thus, all arithmetic performed involving rational expressions mirrors exactly how arithmetic is performed on fractions of numbers.