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]
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]
