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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

Explain how to use long division to write the improper fraction $\frac{172}{5}$ as the sum of an integer and a proper fraction.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{x^{2} - 4} d x$ .

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{\sqrt{x^{2} - 4}} d x$ .

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving $\int \frac{1}{x^{2} + 4} d x .$

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving $\int {(x^{2} + 4)}^{- 5 / 2} d x$

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form $\sqrt{x^{2} - a^{2}}$ .

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with $x = a \sin u$ , we must consider the cases $x > a$ and $x < - a$ separately.

(h) True or False: When using trigonometric substitution with $x = a s e c u$ , we must consider the cases $x > a$ and $x < - a$ separately.

Explain why $\int \frac{2 x}{x^{2} + 1} d x$ and $\int \frac{1}{x \ln x} d x$ are essentially the same integral after a change of variables.

Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial $x^{2} - x + 3$ as the quotient above the top line, and the polynomial 3x − 1 at the bottom as the remainder. Then $p (x) =___a n d \frac{p (x)}{q (x)} =____$

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$\int x \sqrt{x^{2} + 1} d x$

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