# Rational Expressions

**Understanding Rational Expressions**

- A
**Rational Expression**is any expression or equation that can be written as a ratio of two polynomials. - The expression in the numerator and the denominator may be a monomial, binomial or a higher degree polynomial.
- Like a rational number, which has all real numbers as its domain except for zero, a rational expression has all real numbers as its domain except for any value that makes the denominator equal to zero.
- Rational expressions are used in a wide array of problems in algebra, from simple equations to complex real-world word problems.

**Simplifying a Rational Expression**

- To simplify a rational expression, we factorise the numerator and the denominator and cancel out common factors.
- Be cautious, only multiplication allows the cancellation of terms.
**Cancellation is mistaken**when it is tried in addition or subtraction.

**Adding and Subtracting Rational Expressions**

- Before adding or subtracting rational expressions, we must first find a
**common denominator**. - The common denominator is the least common multiple (LCM) of the two denominators.
- Once the common denominator is obtained, rewrite each term as an equivalent fraction with the common denominator. Then add or subtract the numerators of these fractions.

**Multiplying and Dividing Rational Expressions**

- When multiplying rational expressions, we simply multiply the numerators together and the denominators together, and then simplify the result, if possible.
- Rational expressions are divided by multiplying the first expression by the reciprocal of the second.
- As with multiplication, we simplify if possible by cancelling common factors in the numerator and the denominator.

**Evaluating Rational Expressions**

- Provide the value of the variable, substitute this value into the expression and simplify to get the result.
- Ensure the value of the variable is not one which makes the denominator zero - keep in mind the
**domain**of rational expressions.

**Solving Rational Expressions**

- When solving rational equations, multiply every term by the common denominator to eliminate the denominators.
- Remember to check your answers, as this method can introduce extraneous solutions - solutions that do not satisfy the original equation.

**Graphing Rational Expressions**

- When graphing rational functions, it is important to identify any vertical asymptotes and horizontal asymptotes, as well as any potential holes in the graph.
- These are identified by factors that cancel out in your simplified rational expression. A factor that cancels out will be a hole, while a factor that does not cancel will be an asymptote.
- Plot a few points to help complete the sketch, especially at points close to the asymptotes or the hole.