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Polynomial Division

Polynomial Division , Synthetic Division, Rational Expressions

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

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