What is the remainder of the following division problem?
$\begin{array}{l}
\frac{1}{x \longdiv { x + 1 }} \\
\frac{x}{\overline{0+1}}
\end{array}$
Answer (1)
Divide x + 1 by x .
The quotient is 1 and the remainder is 1.
The remainder of the division problem is 1.
The final answer is 1 .
Explanation
Understanding the Problem We are given a polynomial division problem where we divide x + 1 by x . The division is partially completed, showing that x goes into x + 1 one time. The result of subtracting x from x + 1 is 1 . We are asked to find the remainder of this division.
Performing the Division To find the remainder, we perform the polynomial division of x + 1 by x . We can write x + 1 as 1 ⋅ x + 1 . This means that when we divide x + 1 by x , the quotient is 1 and the remainder is 1.
Finding the Remainder Therefore, the remainder of the division problem is 1.
Examples
Polynomial division is used in various applications, such as simplifying complex rational expressions in calculus and engineering. For example, when analyzing circuits, you might encounter a transfer function that is a ratio of polynomials. Using polynomial division, you can simplify this function to better understand the circuit's behavior. Also, polynomial division is used in cryptography, specifically in error-correcting codes, which are essential for reliable data transmission and storage.
