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Chapter 2: Problem 59

How can the Division Algorithm be used to check the quotient and remainder in a long division problem?

Short Answer

Expert verified
By using the Division Algorithm, one can verify the quotient and remainder from a long division operation. Simply use the formula \( a = bq + r \) where \( a \) is the dividend, \( b \) is the divisor, \( q \) is the quotient, and \( r \) is the remainder - your answer from the long division. If this formula holds true, then your quotient and remainder are correct.

Step by step solution

01

Understand the Division Algorithm Theorem

The Division Algorithm states that for any given integers \( a \) and \( b \), where \( b > 0 \), there exist unique integers \( q \) and \( r \) such that \( a = bq + r \) and \( 0 \leq r < b \). The variables in this formula represent the following: \( a \) is the dividend, \( b \) is the divisor, \( q \) is the quotient, and \( r \) is the remainder.
02

Interpret the Division Algorithm in the Context of Long Division

Using long division, you divide \( a \) (the dividend) by \( b \) (the divisor) to get \( q \) (the quotient). There may or may not be a remainder \( r \). So, the relationship between these values if a successful division operation was performed is \( a = bq + r \).
03

Use the Division Algorithm to Check the Quotient and Remainder

To check the quotient and remainder in a long division problem, plug the divisor, quotient, and remainder into \( a = bq + r \) and see if it gives you the dividend. If it does, then you can verify that your long division operation, therefore your quotient and remainder are correct.

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Most popular questions from this chapter

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations of \(f(x)=\frac{1}{x}\) to graph \(g.\) $$g(x)=\frac{3 x-7}{x-2}$$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{5 x^{2}}{x^{2}-4} \cdot \frac{x^{2}+4 x+4}{10 x^{3}}$$

Use a graphing utility to graph \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

Find the horizontal asymptote, if there is one, of the graph of rational function. $$g(x)=\frac{15 x^{2}}{3 x^{2}+1}$$

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{x^{2}}-3$$
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