As you know, a rational number is one that can be expressed as a fraction, that is,

$\frac{p}{q}$,

where $p$ and $q$ are integers (and $q\ne 0$).

Similarly, a **rational expression **(sometimes called an **algebraic fraction**) is one that can be expressed as a quotient of polynomials, i.e. $\frac{p}{q}$
where $p$
and $q$
are polynomials (and $q\ne 0$).

**Example 1:**

$\frac{3}{{x}^{3}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}5{x}^{2}y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}7{y}^{3}}$

is a rational expression, since both the numerator and the denominator are polynomials. ("$3$" counts as a polynomial... it's just a very simple one, with only one term.)

$\frac{5x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}3}{6x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{x}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{x}^{y}}$

is **not** a rational expression. The denominator is **not** a polynomial.

A rational expression can be simplified if the numerator and denominator contain a common factor.

**Example 2:**

Simplify.

$\frac{3x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6}{9{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}9x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}54}$

First, factor out a constant from both numerator and denominator. Write the $9$ as $3\cdot 3$.

$\frac{3x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6}{9{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}9x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}54}=\frac{3x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6}{3\cdot 3({x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}6)}$

Next, factor the quadratic in the denominator. (Look for two numbers with a product of $-6$ and a sum of $-1$.)

$=\frac{3(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2)}{3\cdot 3(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2)(x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}3)}$

Finally, cancel common factors.

$=\frac{1}{3(x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}3)}$

When we factored out $x+2$ in the above expression, we made an important change. The new expression

$\frac{1}{3(x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}3)}$

is defined for $x=-2$; it equals $-\frac{1}{5}$. But the original expression we were trying to simplify,

$\frac{3x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6}{9{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}9x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}54}$

is **undefined** for $x=-2$, because the denominator equals zero (and division by zero is a no-no).

So, our simplification is not really true for all points. When you simplify rational expressions, you should make note of these **excluded values**.