# Rational Expressions and Functions (12) (ELCA)

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## Properties of Rational Expressions and FunctionsEdit

### Rational ExpressionsEdit

**Rational Expressions**– is the quotient or ratio of two polynomials with the denominator not equal to zero.

Example – , , , , and .

### Domain and RangeEdit

**Range**– is a rational set of all real numbers that can be used to find the value of zero. (y) (Dependent Variable) (The Zeros of an equation)**Domain**– is a rational set of all real numbers that can be used in place of the variable. (x) (Independent Variable)

How to Write Sets | ||

Set Builder Notation | {x|x ≠ -5} | x equals all reals except -5 |

Interval Notation | (-∞, -5) U (-5, ∞) |

### Reducing Rational ExpressionsEdit

**Rule of Thirds**–

Strategy for Reducing Rational Expressions |

1) All reducing is done by dividing out common factors. |

2) Factor the numerator and denominator completely to see the common factors. |

3) Use the quotient rule to reduce a ratio of two monomials involving exponents. |

4) We may have to factor out a common factor with a negative sign to get identical factors in the numerator and denominator. |

### FactoringEdit

#### Difference of SquaresEdit

Examples:

#### Difference of CubesEdit

Examples:

#### Sum of Two CubesEdit

#### PolynomialsEdit

Examples:

### Building Up DenominatorEdit

Fractions without identical denominators can be converted to equivalent fractions with a common denominator by reversing the procedure for reducing fractions to the lowest term.

Examples:

### Rational FunctionsEdit

A rational expression can be used to determine the value of a variable.

Examples:

- Find R(3) for

### Real Life Application: Average CostEdit

A car maker spent $700 million to develop a new SUV, which will sell for $40,000. If the cost of manufacturing the SUV is $30,000 each, then what rational function gives the average cost to developing and manufacturing x vehicles? Compare the average cost per vehicle for manufacturing levels of 10,000 vehicles and 100,000 vehicles.

The polynomial gives the cost in dollars for development and manufacture of x vehicles. The average cost per vehicles is:

## Multiplication and DivisionEdit

### Multiplying Rational ExpressionsEdit

One multiplies rational expressions by multiplying their numerators and denominators.

Example:

It is more common to reduce rational expressions *before* multiplying.

Example:

If and are rational numbers, then

More Examples:

### Dividing Rational ExpressionsEdit

When dividing rational numbers, you must multiply by the reciprocal or multiplicative inverse of the divisor.

Example:

Example:

## Addition and SubtractionEdit

In order to add or subtract addition of subtraction problems, one must find a common denominator.

Examples:

## Complex FractionsEdit

**Complex Fractions** – a fraction that has rational expressions in the numerator, the denominational, or both.

Examples: