# Rational Expressions

**• Rational Expression :** A rational expression is the
quotient of two polynomials.

**Example:**

**• Domain of An Algebraic Expression : **The domain of
an algebraic expression is

the set of all numbers for which the expression is defined. (i.e. the set of all
numbers

that can be used in the expression without having a zero in a denominator or a
negative

number under an even radical)

**Example:** The domain of
is all real numbers except -2. Since -2 would cause

the denominator of this expression to be zero, we exclude it from the domain.
All other

real numbers may be substituted without any problems.

**Example:** The domain of
is all real numbers greater than or equal to
1. Since

we cannot take the square root of a negative number, we need x-1 ≥ 0 which
implies

that x ≥ 1.

**Example:** The domain of
can be determined by finding the real numbers

that will cause the denominator to be zero. Therefore, if we factor the
denominator it

will be clear what those numbers are: x^2 - x + 6 = (x - 3)(x + 2). We can see
then,

that 3 and -2 will cause the denominator to be zero. Hence, we exclude these
numbers

from the domain. The domain of the expression is all reals except 3,-2.

**• Simplifying Rational Expressions: **To simplify a
rational expression, follow these

steps:

1. Factor the numerator and denominator completely.

2. Cancel any factor that appears in both the numerator and the denominator.

**Example:**

**• Multiplying Rational Expressions:** This is really
just an extension of simplifying.

The steps are as follows :

1. Factor the numerator and denominator of each rational
expression.

2. Cancel any factor that appears in a numerator and a denominator.

**Example:**

**• Dividing Rational Expressions: **After the first
step, follow the same strategy as in

multiplying rational expressions.

1. Invert (flip over) the rational expression you’re
dividing by (i.e. the second expression

or the one in the denominator of the large fraction )

2. Factor

3. Cancel

**Example:**

**• Adding and Subtracting Rational Expressions: **
Since rational expressions are

basically fractions, in order to be able to add them or subtract them, we must
first

have common denominators , and preferably the LEAST common denominator.

**Strategy:**

1. Factor the denominators.

2. The Least Common Denominator is the product of all the factors present raised

to the highest exponent that appears on the factor in the factorizations.

3. For each fraction, multiply the numerator and denominator by the factors that

are in the LCD , but not in the fraction’s denominator.

4. Add or subtract the numerators as appropriate to the problem.

5. Simplify.

**Example:**

Try these: pp43-44: 4, 10, 16, 34

Answers:

4.) All reals except 1,-1,-2

10.) 1

16.) a - b

Prev | Next |