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Chapter 3: Problem 48

In your own words, state the Division Algorithm.

Short Answer

Expert verified
The Division Algorithm states that for given two integers 'a' and 'b' with 'b' not equal to zero, there exist unique integers 'q' and 'r' such that 'a' equals to 'b' times 'q' plus 'r' where 'r' is greater or equal to 0 and less than 'b'.

Step by step solution

01

Understanding the Division Algorithm

The Division Algorithm states: For any given integers a and b, with b > 0, there exist unique integers q and r satisfying the equation \(a = bq + r\) where \(0 \leq r < b\). This theorem basically is describing the process of long division.
02

Describing the Division

In the equation \(a = bq + r\), 'a' is the dividend, 'b' is the divisor, 'q' is the quotient and 'r' is the remainder. When we divide 'a' by 'b', we get a quotient 'q' and a remainder 'r'.
03

Discussing Uniqueness and Range of Remainder

The Division Algorithm also tells us two things about the quotient and the remainder: Firstly, the quotient 'q' and the remainder 'r' are unique for given 'a' and 'b'. Secondly, the remainder 'r' always satisfy the property that \(0 \leq r < b\), meaning the remainder yielded from division will always be a non-negative integer and less than the divisor 'b'.

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Most popular questions from this chapter

What is a polynomial function?

Which one of the following is true? a. The graph of a rational function cannot have both a vertical and a horizontal asymptote. b. It is not possible to have a rational function whose graph has no \(y\) -intercept. c. The graph of a rational function can have three horizontal asymptotes. d. The graph of a rational function can never cross a vertical asymptote.

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(58-61 .\) Then determine the number of real zeros and the number of nonreal complex zeros for each function. $$ f(x)=3 x^{4}+4 x^{3}-7 x^{2}-2 x-3 $$

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(58-61 .\) Then determine the number of real zeros and the number of nonreal complex zeros for each function. $$ f(x)=x^{3}-6 x-9 $$

Explain why nonreal complex zeros are gained or lost in pairs in terms of graphs of polynomial functions.
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