College

What dividend is represented by the synthetic division below?

\[
\begin{array}{c|ccc}
5 & 2 & 10 & 1 \\
& & -10 & 0 \\
\hline
& 2 & 0 & 1
\end{array}
\]

The polynomial represented by the synthetic division is:

\[
2x^3 + 10x^2 + x + 5
\]

Answer :

We start with the synthetic division table where the left number is $5$, and the coefficients of the dividend are written as:

$$
\begin{array}{cccc}
5\,| & 2 & 10 & 1 \\
& & -10 & 0 \\
\hline
& 2 & 0 & 1 \\
\end{array}
$$

Even though the number on the left is $5$, the multiplication step uses $-5$ (since $2 \times (-5) = -10$). This indicates that the divisor is actually

$$
x - (-5) = x + 5.
$$

Now, follow these steps:

1. **Bring down the first coefficient:**
The first coefficient is $2$, so it remains unchanged.

2. **Multiply by the divisor’s associated number:**
Multiply $2$ by $-5$ to obtain $-10$.

3. **Add to the next coefficient:**
Add $-10$ to the second coefficient $10$, resulting in $0$.

4. **Multiply again by $-5$:**
Multiply the result $0$ by $-5$ to get $0$.

5. **Add to the last coefficient:**
Add $0$ to the third coefficient $1$, resulting in a remainder of $1$.

The bottom row now shows the quotient coefficients and the remainder:

- **Quotient coefficients:** $[2, 0]$, which represents the quotient polynomial $2x + 0$ (simply $2x$).
- **Remainder:** $1$.

By the polynomial division algorithm, we have:

$$
\text{Dividend} = (\text{Divisor})(\text{Quotient}) + \text{Remainder}.
$$

Substitute the known values:

$$
\text{Dividend} = (x + 5)(2x) + 1.
$$

Now, expand the product:

$$
(x + 5)(2x) = 2x^2 + 10x.
$$

Finally, adding the remainder:

$$
2x^2 + 10x + 1.
$$

Thus, the dividend polynomial is:

$$
\boxed{2x^2 + 10x + 1}.
$$