High School

In the synthetic-division problem shown below, what number belongs in the place of the question mark?

[tex]
\[
\begin{array}{r|lll}
-2 & 1 & -2 & -8 \\
& & -2 & \square & 5 \\
\hline
& 1 & -4 & \square \\
\end{array}
\]
[/tex]

Answer :

Sure! Let's work through the synthetic division step by step.

We are dividing a polynomial using synthetic division, with the given setup:

```
-2 | 1 -2 -8
| ?
--------------
1 -4 ?
```

Here's how synthetic division works:

1. Bring down the first coefficient: The first number from the top row (which is 1) is simply brought down to the bottom row. This becomes the first number in the second row.

2. Multiply and add:
- Multiply this number (1) by the divisor (-2). So, 1 -2 = -2.
- Add this result to the next number in the top row (-2). So, -2 + (-2) = -4. Fill this in the next position in the bottom row.

3. Repeat the process:
- Take the number you just wrote (-4) and multiply it by the divisor (-2). So, -4 -2 = 8.
- Add this result to the next number in the top row (-8). So, 8 + (-8) = 0. Fill this in the last position of the bottom row.

Therefore, the numbers that belong in place of the question marks are:
- The first question mark in the middle row is -4.
- The last question mark in the bottom row is 0.

So, the complete filled synthetic division setup is:

```
-2 | 1 -2 -8
| -4
--------------
1 -4 0
```

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