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Chapter 2: Problem 1

Two forms of the Division Algorithm are shown below. Identify and label each term or function. \(f(x)=d(x) q(x)+r(x) \quad \frac{f(x)}{d(x)}=q(x)+\frac{r(x)}{d(x)}\)

Short Answer

Expert verified
In both division algorithms, \(f(x)\) is the dividend, \(d(x)\) is the divisor, \(q(x)\) is the quotient, and \(r(x)\) is the remainder. In the second formula, \(\frac{r(x)}{d(x)}\) is the ratio of the remainder to the divisor.

Step by step solution

01

Understanding the equations

We have two forms of the Division Algorithm here.\nThe first form is \(f(x)=d(x) q(x)+r(x)\) and the second is \(\frac{f(x)}{d(x)}=q(x)+\frac{r(x)}{d(x)}\). These formulas show how polynomial division works.
02

Identify and label each term or function in the first formula

Looking at the first equation \(f(x)=d(x) q(x)+r(x)\), we can say that: \n1. \(f(x)\) is the dividend, the polynomial to be divided.\n2. \(d(x)\) is the divisor, the polynomial we are dividing by. \n3. \(q(x)\) is the quotient, the result of the division.\n4. \(r(x)\) is the remainder, what is left over after division.
03

Identify and label each term or function in the second formula

In the second division algorithm \(\frac{f(x)}{d(x)}=q(x)+\frac{r(x)}{d(x)}\), we still have: \n1. \(f(x)\) as the dividend.\n2. \(d(x)\) as the divisor.\n3. \(q(x)\) as the quotient.\n4. \(r(x)\) as the remainder. Additionally, the \(\frac{r(x)}{d(x)}\) term represents the ratio of the remainder to the divisor.

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