Long Division

**Check: **Dividend = (Quotient)(Divisor) + Remainder

** Dividing by a monomial :**

**Dividing Two Polynomials with more than One Term:**

(1) Write terms in **each polynomial** in descending

order according to degree.

(2) Insert missing terms in **both polynomials** with a 0

coefficient.

(3) Use Long Division algorithm. The remainder is a

polynomial whose degree is less than the degree of the

divisor.

**Example:** Perform the division.

**Synthetic Division**

Synthetic division is used when a polynomial is divided

by a first-degree binomial of the form x − k .

← Coefficients of Dividend

**Diagonal pattern: **Multiply by k

**Vertical pattern: **Add terms

**Example: **Use synthetic division to find the quotient

and remainder.

**Example: **Verify that x − 3 is a factor of

**Rational Expressions**

A **rational expression** is a ratio of two polynomials.

The **domain** of an expression in one variable is the set

of all real numbers for which the expression is defined.

The **domain of a rational expression** is the set of all real

numbers that do not make the denominator equal to

zero .

** Reducing Rational Expressions:**

**Example:** Find the domain and reduce the expression
to

lowest terms .

Domain:

Domain:

** Multiplication and division :**

**Example: **Perform the indicated operations and

simplify. Give restrictions on the variables .

**Addition and subtraction:**

In order to add/ subtract rational expressions we use the

Least Common Multiple **(LCM)** of the denominators.

**To Find the LCM of the Denominators:**

1. Factor polynomials that are in the denominators.

2. The **LCM** is the product of all different factors

which are in the denominators (numbers, variables,

expressions) each raised to the largest power that

appears on that factor.

**Example:** Add or subtract, as indicated. Give all

restrictions on the variables.

**Note:** Be aware of the case when the denominators

are additive inverses of each other.

**Example: **Perform the indicated operations.

**Mixed Quotients**

A mixed quotient (complex fraction) is a quotient of

rational expressions.

Simplifying a Complex Fraction:

** Method 1:** Multiply both numerator and denominator

by the LCM of the denominators of all

simple fractions and simplify .

**Method 2: **Perform the indicated additions and/or

subtractions in the numerator and

denominator and then divide.

**Example: **Simplify the complex fraction (mixed

quotient).

**Example: **Write the fraction as a sum of two or more

expressions.

**Example:** Perform the indicated operations and
simplify