TBIL-Precal Polynomial Long Division (PR4)

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Section 4.4 Polynomial Long Division (PR4)

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Objectives

Rewrite a rational function as a polynomial plus a proper rational function.

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Subsection 4.4.1 Activities

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Observation 4.4.1.

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We have seen previously that we can reduce rational functions by factoring, for example

\frac{x^{2} + 5 x + 4}{x^{3} + 3 x^{2} + 2 x} = \frac{(x + 1) (x + 4)}{x (x + 2) (x + 1)} = \frac{x + 4}{x (x + 2)} .

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In this section, we will explore the question: what can we do to simplify rational functions if we are not able to reduce by easily factoring?

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Definition 4.4.2.

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Recall that a fraction is called proper if its numerator is smaller than its denominator, and improper if the numerator is larger than the denominator (so

\frac{3}{5}

is a proper fraction, but

\frac{32}{7}

is an improper fraction). Similarly, we define a proper rational function to be a rational function where the degree of the numerator is less than the degree of the denominator.

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Activity 4.4.3.

Label each of the following rational functions as either proper or improper.

$\frac{x^{3} + x}{x^{2} + 4}$
$\frac{3}{x^{2} + 3 x + 4}$
$\frac{7 + x^{3}}{x^{2} + x + 1}$
$\frac{x^{4} + x + 1}{x^{4} + 4 x^{2}}$

Answer.

A, C, and D are improper, while B is proper.

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Observation 4.4.4.

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When dealing with an improper fraction such as

\frac{32}{7},

it is sometimes useful to rewrite this as an integer plus a proper fraction, e.g.

\frac{32}{7} = 4 + \frac{4}{7} .

Similarly, it will sometimes be useful to rewrite an improper rational function as the sum of a polynomial and a proper rational function, such as

\frac{x^{3} + x}{x^{2} + 4} = x - \frac{3 x}{x^{2} + 4} .

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Activity 4.4.5.

Consider the improper fraction

\frac{357}{11} .

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(a)

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Use long division to write

\frac{357}{11}

as an integer plus a proper fraction.

Answer.

\frac{357}{11} = 32 + \frac{5}{11} .

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(b)

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Now we will carefully redo this process in a way that we can generalize to rational functions. Note that we can rewrite

357

357 = 3 \cdot 10^{2} + 5 \cdot 10 + 7,

and

11

11 = 1 \cdot 10 + 1 .

By comparing the leading terms in these expansions, we see that to knock off the leading term of

357,

we need to multiply

11

3 \cdot 10^{1} .

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Using the fact that

357 = 11 \cdot 30 + 27,

rewrite

\frac{357}{11}

\frac{357}{11} = 30 + \frac{?}{11} .

Answer.

\frac{357}{11} = 30 + \frac{27}{11} .

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(c)

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Note now that if we can rewrite

\frac{27}{11}

as an integer plus a proper fraction, we will be done, since

\frac{357}{11} = 30 + \frac{27}{11} .

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Rewrite

\frac{27}{11} = ? + \frac{?}{11}

as an integer plus a proper fraction.

Answer.

\frac{27}{11} = 2 + \frac{5}{11} .

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(d)

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Combine your work in the previous two parts to rewrite

\frac{357}{11}

as an integer plus a proper fraction. How does this compare to what you obtained in part (a)?

Answer.

\frac{357}{11} = 30 + \frac{27}{11} = 30 + 2 + \frac{5}{11} = 32 + \frac{5}{11} .

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Activity 4.4.6.

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Now let’s consider the rational function

\frac{3 x^{2} + 5 x + 7}{x + 1} .

We want to rewrite this as a polynomial plus a proper rational function.

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(a)

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Looking at the leading terms, what do we need to multiply

x + 1

by so that it would have the same leading term as

3 x^{2} + 5 x + 7 ?

$3$
$x$
$3 x$
$3 x + 5$

Answer.

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(b)

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Rewrite

3 x^{2} + 5 x + 7 = 3 x (x + 1) + ?,

and use this to rewrite

\frac{3 x^{2} + 5 x + 7}{x + 1} = 3 x + \frac{?}{x + 1} .

Answer.

\frac{3 x^{2} + 5 x + 7}{x + 1} = 3 x + \frac{2 x + 7}{x + 1}

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(c)

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Now focusing on

\frac{2 x + 7}{x + 1},

what do we need to multiply

x + 1

by so that it would have the same leading term as

2 x + 7 ?

$2$
$x$
$2 x$
$2 x + 7$

Answer.

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(d)

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Rewrite

\frac{2 x + 7}{x + 1} = 2 + \frac{?}{x + 1} .

Answer.

\frac{2 x + 7}{x + 1} = 2 + \frac{5}{x + 1}

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(e)

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Combine this with the previous parts to rewrite

\frac{3 x^{2} + 5 x + 7}{x + 1} = 3 x + ? + \frac{?}{x + 1} .

Answer.

\frac{3 x^{2} + 5 x + 7}{x + 1} = 3 x + 2 + \frac{5}{x + 1}

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Activity 4.4.7.

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Next we will use the notation of long division to rewrite the rational function

\frac{3 x^{2} + 5 x + 7}{x + 1}

as a polynomial plus a proper rational function.

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(a)

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First, let’s use long division notation to write the quotient.

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What do we need to multiply

x + 1

by so that it would have the same leading term as

3 x^{2} + 5 x + 7 ?

Answer.

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(b)

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Now to rewrite

3 x^{2} + 5 x + 7

3 x (x + 1) + ?,

place the product

3 x (x + 1)

below and subtract.

Answer.

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(c)

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Now focusing on

2 x + 7,

what do we need to multiply

x + 1

by so that it would have the same leading term as

2 x + 7 ?

Answer.

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(d)

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Now, subtract

2 (x + 1)

to finish the long division.

Answer.

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(e)

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This long division calculation has shown that

3 x^{2} + 5 x + 7 = (x + 1) (3 x + 2) + 5 .

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Use this to rewrite

\frac{3 x^{2} + 5 x + 7}{x + 1}

as a polynomial plus a proper rational function.

Answer.

\frac{3 x^{2} + 5 x + 7}{x + 1} = 3 x + 2 + \frac{5}{x + 1}

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Remark 4.4.8.

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Note that in Activity 4.4.6 and Activity 4.4.7 we performed the same computations, but just organized our work a little differently.

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Activity 4.4.9.

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Rewrite

\frac{x^{2} + 1}{x - 1}

as a polynomial plus a proper rational function.

Hint.

Note that

x^{2} + 1 = x^{2} + 0 x + 1 .

Answer.

x + 1 + \frac{3}{x - 1} .

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Activity 4.4.10.

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Rewrite

\frac{x^{5} + x^{3} + 2 x^{2} - 6 x + 7}{x^{2} + x - 1}

as a polynomial plus a proper rational function.

Answer.

x^{3} - x^{2} + 3 x - 2 + \frac{- x + 5}{x^{2} + x - 1} .

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Activity 4.4.11.

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Rewrite

\frac{3 x^{4} - 5 x^{2} + 2}{x - 1}

as a polynomial plus a proper rational function.

Answer.

3 x^{3} + 3 x^{2} - 2 x - 2 .

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Exercises 4.4.2 Exercises

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Exercises available at https://tbil.org/precalculus/preview/exercises/#/bank/PR4/.

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