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Chapter 5: Q. 6 (page 441)

Describe two ways in which the long-division algorithm for polynomials is similar to the long-division algorithm for integers and then two ways in which the two algorithms are different.

Short Answer

Expert verified

The long division algorithm to divide polynomial is analogous to the long division algorithm for integers.

Step by step solution

01

Step 1. Given Information

The given term is long-division algorithm for polynomials

02

Step 2. Explanation

Any two numbers can be divided as long as the divisor is not equal to 0.

similarly any two polynomial can be divided as long as the divisor is not equal to 0.

The long division algorithm to divide polynomial is analogous to the long division algorithm for integers.

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Most popular questions from this chapter

Show that if $x = \tan u$ , then $d x = \sec^{2} u d u$ , in the following two ways: (a) by using implicit differentiation, thinking of $u$ as a function of $x$ , and (b) by thinking of $x$ as a function of $u$ .

Complete the square for each quadratic in Exercises 28–33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

$x^{2} - 4 x - 8$

Find three integrals in Exercises 27–70 for which either algebra or u-substitution is a better strategy than integration by parts.

True/False: Determinewhethereachofthestatementsthat follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: $f (x) = \frac{x + 1}{x - 1}$ is a proper rational function.

(b) True or False: Every improper rational function can be expressed as the sum of a polynomial and a proper rational function.

(c) True or False: After polynomial long division of p(x) by q(x), the remainder r(x) has a degree strictly less than the degree of q(x).

(d) True or False: Polynomial long division can be used to divide two polynomials of the same degree.

(e) True or False: If a rational function is improper, then polynomial long division must be applied before using the method of partial fractions.

(f) True or False: The partial-fraction decomposition of $\frac{x^{2} + 1}{x^{2} (x - 3)}$ is of the form $\frac{A}{x^{2}} + \frac{B}{x - 3}$

(g) True or False: The partial-fraction decomposition of $\frac{x^{2} + 1}{x^{2} (x - 3)}$ is of the form $\frac{B x + C}{x^{2}} + \frac{A}{x - 3}$ .

(h) True or False: Every quadratic function can be written in the form $A {(x - k)}^{2} + C$

Solve $\int \frac{x^{2}}{1 + x^{2}} d x$ the following two ways:

(a) with the substitution $u = \tan^{- 1} x;$

(b) with the trigonometric substitution x = tan u.

See all solutions

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