Polynomial Long Division

An Example.

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

displaymath238

in the form

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The expression

displaymath240

is called the quotient, the expression

displaymath241

is called the divisor and the term

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is called the remainder. What is special about the way the expression above is written? The remainder 28x+30 has degree 1, and is thus less than the degree of the divisor tex2html_wrap_inline272 .

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

DIVISION ALGORITHM: If f(x) and tex2html_wrap_inline276 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

displaymath243

and so that the degree of r(x) is less than the degree of d(x). In the special case where r(x)=0, we say that d(x) divides evenly into f(x).

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

displaymath238

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

displaymath245

First divide the leading term tex2html_wrap_inline296 of the numerator polynomial by the leading term tex2html_wrap_inline298 of the divisor, and write the answer 3x on the top line:

displaymath246

Now multiply this term 3x by the divisor tex2html_wrap_inline272 , and write the answer

displaymath247

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

displaymath248

Next subtract the last line from the line above it:

displaymath249

Now repeat the procedure: Divide the leading term tex2html_wrap_inline306 of the polynomial on the last line by the leading term tex2html_wrap_inline298 of the divisor to obtain -11, and add this term to the 3x on the top line:

displaymath250

Then multiply "back": tex2html_wrap_inline314 and write the answer under the last line polynomial, lining up terms of equal degree:

displaymath251

Subtract the last line from the line above it:

displaymath252

You are done! (In the next step, you would divide 28x by tex2html_wrap_inline298 , not yielding a polynomial expression!) The remainder is the last line: 28x+30, and the quotient is the expression on the very top: 3x-11. Consequently,

displaymath253


How to check your answer?

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

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then to multiply out:

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and then to simplify the right side:

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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

displaymath330

Write the expression in a form reminiscent of long division:

displaymath331

First divide the leading term tex2html_wrap_inline356 of the numerator polynomial by the leading term x of the divisor, and write the answer tex2html_wrap_inline298 on the top line:

displaymath332

Now multiply this term tex2html_wrap_inline298 by the divisor x+2, and write the answer

displaymath333

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

displaymath334

Next subtract the last line from the line above it:

displaymath335

Now repeat the procedure: Divide the leading term tex2html_wrap_inline366 of the polynomial on the last line by the leading term x of the divisor to obtain -2x, and add this term to the tex2html_wrap_inline298 on the top line:

displaymath336

Then multiply "back": tex2html_wrap_inline374 and write the answer under the last line polynomial, lining up terms of equal degree:

displaymath337

Subtract the last line from the line above it:

displaymath338

You have to repeat the procedure one more time.

Divide:

displaymath339

Multiply "back":

displaymath340

and subtract:

displaymath341

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: tex2html_wrap_inline382 . Consequently,

displaymath342


An Example: Long Polynomial Division and Factoring.

Let's use polynomial long division to rewrite

displaymath384

Write the expression in a form reminiscent of long division:

displaymath385

First divide the leading term tex2html_wrap_inline356 of the numerator polynomial by the leading term tex2html_wrap_inline298 of the divisor, and write the answer x on the top line:

displaymath386

Now multiply this term x by the divisor tex2html_wrap_inline414 , and write the answer

displaymath387

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

displaymath388

Next subtract the last line from the line above it:

displaymath389

Now repeat the procedure: Divide the leading term tex2html_wrap_inline416 of the polynomial on the last line by the leading term tex2html_wrap_inline298 of the divisor to obtain -5, and add this term to the x on the top line:

displaymath390

Then multiply "back": tex2html_wrap_inline424 and write the answer under the last line polynomial, lining up terms of equal degree:

displaymath391

Subtract the last line from the line above it:

displaymath392

You are done! In this case, the remainder is 0, so tex2html_wrap_inline414 divides evenly into tex2html_wrap_inline430 .

Consequently,

displaymath393

Multiplying both sides by the divisor tex2html_wrap_inline414 yields:

displaymath394

In this case, we have factored the polynomial tex2html_wrap_inline430 , i.e., we have written it as a product of two "easier" (=lower degree) polynomials.


Exercise 1.

Use long polynomial division to rewrite

displaymath436

Answer.

Exercise 2.

Use long polynomial division to rewrite

displaymath438

Answer.

Exercise 3.

Use long polynomial division to rewrite

displaymath440

Answer.

Exercise 4.

Use long polynomial division to rewrite

displaymath442

Answer.

Exercise 5.

Use long polynomial division to rewrite

displaymath444

Answer.

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Helmut Knaust
Fri Jun 6 13:11:33 MDT 1997

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