To solve the polynomial [tex]\((x^3 - x^2 - 17x - 15)\)[/tex] divided by [tex]\((x - 5)\)[/tex] using synthetic division, we will follow the step-by-step procedure:
1. Setup: Write down the coefficients of the polynomial [tex]\(x^3 - x^2 - 17x - 15\)[/tex]. They are: [tex]\([1, -1, -17, -15]\)[/tex].
2. Root Identification: Identify the root provided by the divisor [tex]\((x - 5)\)[/tex]. The root is [tex]\(5\)[/tex].
3. Synthetic Division Process:
- Write the root (5) on the left.
- Write the coefficients of the polynomial in order on the right.
```
5 | 1 -1 -17 -15
-----------------
```
4. Perform Synthetic Division Steps:
- Bring down the first coefficient (1) as is.
```
5 | 1 -1 -17 -15
-----------------
1
```
- Multiply the root (5) by the value just written below the line (1), resulting in [tex]\(5 \times 1 = 5\)[/tex].
- Write this result under the next coefficient [tex]\((-1)\)[/tex] and add them [tex]\((-1 + 5 = 4)\)[/tex].
```
5 | 1 -1 -17 -15
-----------------
1 4
```
- Repeat the process: multiply the root (5) by the value just written below the line (4), giving [tex]\(5 \times 4 = 20\)[/tex].
- Write this result under the next coefficient [tex]\((-17)\)[/tex] and add them [tex]\((-17 + 20 = 3)\)[/tex].
```
5 | 1 -1 -17 -15
-----------------
1 4 3
```
- Repeat the process: multiply the root (5) by the value just written below the line (3), resulting in [tex]\(5 \times 3 = 15\)[/tex].
- Write this result under the next coefficient [tex]\((-15)\)[/tex] and add them [tex]\((-15 + 15 = 0)\)[/tex].
```
5 | 1 -1 -17 -15
-----------------
1 4 3 0
```
5. Interpret Results:
- The values below the line [tex]\([1, 4, 3, 0]\)[/tex] represent the coefficients of the quotient polynomial and the remainder.
- The leading coefficient of the quotient is 1, so the quotient polynomial is [tex]\(x^2 + 4x + 3\)[/tex]. The remainder is 0.
Thus, the quotient of the division [tex]\((x^3 - x^2 - 17x - 15)\)[/tex] by [tex]\((x - 5)\)[/tex] is [tex]\(\boxed{x^2 + 4x + 3}\)[/tex].
