Table of Contents

Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 8: Dividing polynomials

**Topic**. This lesson covers Session 8: Dividing polynomials

**Learning Outcomes.**

- Use the long division algorithm to divide two polynomials, determining the quotient and remainder,
- Understand the connection between long division, factors, and roots, and use this connection to solve problems.

**WeBWorK**. There is one WeBWorK assignments on today’s material:

- Polynomials – Division

**Additional Video Resources.**

**Question of the Day: **When we divide two numbers, the result is a number. When we divide two polynomials, what is the result?

## Introduction to Polynomials

**Definition**. A **monomial** is a number, a variable, or a product of numbers and variables. A **polynomial** is a sum (or difference) of monomials.

**Example**. The following are examples of monomials:

The following are examples of polynomials:

In particular, every monomial is also a polynomial.

In this class, we will focus on polynomials with just one variable — we will consider these as functions. For example is such a polynomial.

**Definition**. A **polynomial** is a function of the form

- The real numbers are called
**coefficients**(sometimes we allow the coefficients to be complex numbers as well). - The domain of a polynomial is all real numbers (sometimes we allow the domain to include complex numbers as well).
- The coefficient in front of the highest power of x is called the
**leading coefficient**and is the**degree**of the polynomial. - The zeros of a polynomial are usually referred to as
**roots**. Therefore is a root of a polynomial precisely when .

**Definition**. A **rational function** is a fraction of two polynomials where and are both polynomials. The domain of is all real numbers for which the denominator is not zero:

**Example**. The following are examples of rational functions:

## Long Division

We now show how to divide two polynomials. The method is very similar to the long division of natural numbers.

**Example 8.6**. Divide the following fractions via long division:

a)

b)

*VIDEO: Example – Long division of Natural Numbers and Polynomials*

When we divide 3571 by 11 we get a result of 324 remainder 7. We can express this in two different ways:

In division problems, we have names for all these numbers:

- 3571 is the
**dividend**(the thing we are dividing into) - 11 is the
**divisor**(the thing we are dividing by) - 324 is the
**quotient**(the number of times it goes in) - 7 is the
**remainder**(whatever is left)

When we divide by we get a result of and a remainder of . We can express this in two different ways:

In polynomial division problems, we use the same names for each item — but in this case, they are functions instead of numbers:

- is the
**dividend**(the thing we are dividing into) - is the
**divisor**(the thing we are dividing by) - is the
**quotient**(the number of times it goes in) - is the
**remainder**(whatever is left). NOTE: The divisor will always be a polynomial with degree*less than*the degree of the divisor.

## Dividing by

We now turn our attention to a particular example of division, where the **divisor** has the form . Why do we care about this form in particular? It turns out that dividing by has a nice connection to the **roots**, or **zeros**, of a polynomial.

Observation. Suppose we divide a polynomial by a polynomial , and the result has remainder :

Now, let us plug the value into both sides of this equation:

Notice that the remainder is exactly equal to what we get when we evaluate the function at (we call this fact the **remainder theorem**):

The reminder when dividing by is .

Thinking about what happens when we get a remainder of , we come up with the **factor theorem** — it tells us that the polynomial will go into evenly (no remainder) exactly when .

**Example.**

a) Find the remainder of dividing by

b) Determine whether is a factor of

c) Show that 5 is a root of , and use this to factor *completely*.

*VIDEO: Factor Theorem and Remainder Theorem examples*

#### Exit Question

Divide by long division and use the result to rewrite