## Polynomial Long Division

An Example.

In this section you will learn how to rewrite a rational function such as

in the form

The expression

is called the **quotient**, the expression

is called the **divisor** and the term

is called the **remainder**. What is special about the way the expression above is written? The remainder 28*x*+30 has degree 1, and is thus less than the degree of the divisor .

It is always possible to rewrite a rational function in this manner:

DIVISION ALGORITHM: If f(x) and are polynomials, and the degree of d(x) is less than or equal to the degree of f(x), then there exist unique polynomials q(x) and r(x), so that and so that the degree of |

How do you do this? Let's look at our example

in more detail. Write the expression in a form reminiscent of long division:

First divide the leading term of the numerator polynomial by the leading term of the divisor, and write the answer 3*x* on the top line:

Now multiply this term 3*x* by the divisor , and write the answer

under the numerator polynomial, lining up terms of equal degree:

Next subtract the last line from the line above it:

Now repeat the procedure: Divide the leading term of the polynomial on the last line by the leading term of the divisor to obtain -11, and add this term to the 3*x* on the top line:

Then multiply "back": and write the answer under the last line polynomial, lining up terms of equal degree:

Subtract the last line from the line above it:

You are done! (In the next step, you would divide 28*x* by , not yielding a polynomial expression!) The remainder is the last line: 28*x*+30, and the quotient is the expression on the very top: 3*x*-11. Consequently,

How to check your answer?

The easiest way to check your answer algebraically is to multiply both sides by the divisor:

then to multiply out:

and then to simplify the right side:

Indeed, both sides are equal! Other ways of checking include graphing both sides (if you have a graphing calculator), or plugging in a few numbers on both sides (this is not always 100% foolproof).

Another Example.

Let's use polynomial long division to rewrite

Write the expression in a form reminiscent of long division:

First divide the leading term of the numerator polynomial by the leading term *x* of the divisor, and write the answer on the top line:

Now multiply this term by the divisor *x*+2, and write the answer

under the numerator polynomial, carefully lining up terms of equal degree:

Next subtract the last line from the line above it:

Now repeat the procedure: Divide the leading term of the polynomial on the last line by the leading term *x* of the divisor to obtain -2*x*, and add this term to the on the top line:

Then multiply "back": and write the answer under the last line polynomial, lining up terms of equal degree:

Subtract the last line from the line above it:

You have to repeat the procedure one more time.

Divide:

Multiply "back":

and subtract:

You are done! (In the next step, you would divide -9 by *x*, not yielding a polynomial expression!) The remainder is the last line: -9 (of degree 0), and the quotient is the expression on the very top: . Consequently,

#### An Example: Long Polynomial Division and Factoring.

Let's use polynomial long division to rewrite

Write the expression in a form reminiscent of long division:

First divide the leading term of the numerator polynomial by the leading term of the divisor, and write the answer *x* on the top line:

Now multiply this term *x* by the divisor , and write the answer

under the numerator polynomial, carefully lining up terms of equal degree:

Next subtract the last line from the line above it:

Now repeat the procedure: Divide the leading term of the polynomial on the last line by the leading term of the divisor to obtain -5, and add this term to the *x* on the top line:

Then multiply "back": and write the answer under the last line polynomial, lining up terms of equal degree:

Subtract the last line from the line above it:

You are done! In this case, the remainder is 0, so divides evenly into .

Consequently,

Multiplying both sides by the divisor yields:

In this case, we have **factored** the polynomial , i.e., we have written it as a product of two "easier" (=lower degree) polynomials.