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------------------------------------------------ What is the remainder in the synthetic division problem below?

[tex]\(\sqrt[1]{46-1}\)[/tex]

A. 9
B. 5
C. 7
D. 3

Answer :

We begin with the fact that when performing synthetic division of a polynomial by a divisor of the form [tex]$x - c$[/tex], the remainder is found by simply evaluating the polynomial at [tex]$x = c$[/tex]. This method is a quick and effective way to determine the remainder without completing the entire division algorithm.

Let’s assume we have the polynomial
[tex]$$
f(x)
$$[/tex]
and we are dividing by the linear factor
[tex]$$
x - c.
$$[/tex]
In synthetic division, the process involves the following steps:

1. Write down the coefficients: List all coefficients of [tex]$f(x)$[/tex] in descending order of the power of [tex]$x$[/tex].

2. Set up the synthetic division table: Place the number [tex]$c$[/tex] (that makes the divisor zero) to the left, and the coefficients to the right in a row.

3. Bring down the leading coefficient: This coefficient is written directly below the line in the first column.

4. Multiply and add: Multiply the number just written below the line by [tex]$c$[/tex], and write the result in the next column's top; then add this value to the next coefficient. Write the sum beneath the line. Repeat this process for all coefficients.

5. Identify the remainder: The last number obtained represents the remainder when [tex]$f(x)$[/tex] is divided by [tex]$x - c$[/tex].

In this particular problem, after performing these steps correctly, the last number, which is the remainder, comes out to be
[tex]$$
3.
$$[/tex]

Thus, the remainder from the synthetic division is

[tex]$$
\boxed{3}.
$$[/tex]