Answer :

To determine whether [tex]\( x+2 \)[/tex] is a factor of [tex]\( 4x^6 + 38x^3 + 48 \)[/tex] using synthetic division, follow these steps:

1. Write down the coefficients:
The polynomial [tex]\( 4x^6 + 38x^3 + 48 \)[/tex] can be expressed with the coefficients of all terms including those with zeros:
[tex]\[
4x^6 + 0x^5 + 0x^4 + 38x^3 + 0x^2 + 0x + 48
\][/tex]
This gives us the coefficients: [tex]\( [4, 0, 0, 38, 0, 0, 48] \)[/tex].

2. Set up for synthetic division:
We are dividing by [tex]\( x + 2 \)[/tex], which means we use [tex]\( -2 \)[/tex] in the synthetic division.

3. Perform synthetic division:
- Write [tex]\( -2 \)[/tex] on the left.
- Write the coefficients [tex]\( 4, 0, 0, 38, 0, 0, 48 \)[/tex] to the right.

```
-2 | 4 0 0 38 0 0 48
```

4. Begin the synthetic division:
- Bring down the first coefficient (4) to the bottom row.

```
| 4 0 0 38 0 0 48
-------------------------
| 4
```

- Multiply [tex]\( 4 \)[/tex] by [tex]\( -2 \)[/tex] and write the result below the next coefficient [tex]\( 0 \)[/tex].

```
| 4 0 0 38 0 0 48
| -8
-------------------------
| 4 -8
```

- Add the next coefficient (0) to [tex]\( -8 \)[/tex] and write the result below.

```
| 4 0 0 38 0 0 48
| -8 16
-------------------------
| 4 -8 16
```

- Repeat this process:

- Multiply [tex]\( 16 \)[/tex] by [tex]\( -2 \)[/tex] to get [tex]\( -32 \)[/tex] and add to [tex]\( 38 \)[/tex] to get [tex]\( 6 \)[/tex].
- Multiply [tex]\( 6 \)[/tex] by [tex]\( -2 \)[/tex] to get [tex]\( -12 \)[/tex] and add to [tex]\( 0 \)[/tex] to get [tex]\( -12 \)[/tex].
- Multiply [tex]\( -12 \)[/tex] by [tex]\( -2 \)[/tex] to get [tex]\( 24 \)[/tex] and add to [tex]\( 0 \)[/tex] to get [tex]\( 24 \)[/tex].
- Multiply [tex]\( 24 \)[/tex] by [tex]\( -2 \)[/tex] to get [tex]\( -48 \)[/tex] and add to [tex]\( 48 \)[/tex] to get [tex]\( 0 \)[/tex].

Continue these steps to complete the table:

```
| 4 0 0 38 0 0 48
| -8 16 -32 6 -12 24 -48
-------------------------
| 4 -8 16 6 -12 24 0
```

5. Interpret the result:
The remainder is the last number in the bottom row, which is 0. The division results in a quotient of [tex]\( 4x^5 - 8x^4 + 16x^3 + 6x^2 - 12x + 24 \)[/tex] and a remainder of 0.

Since the remainder is 0, [tex]\( x + 2 \)[/tex] is indeed a factor of [tex]\( 4x^6 + 38x^3 + 48 \)[/tex].

So, [tex]\( x+2 \)[/tex] is a factor of the polynomial [tex]\( 4x^6 + 38x^3 + 48 \)[/tex].

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Rewritten by : Barada