Sure, let's use synthetic division to divide the polynomial [tex]\( \frac{3x^3 + 20x^2 + 22x - 8}{x + 5} \)[/tex].
### Step-by-Step Process:
1. Identify the coefficients of the polynomial: The given polynomial is [tex]\(3x^3 + 20x^2 + 22x - 8\)[/tex]. The coefficients are [tex]\(3, 20, 22, -8\)[/tex].
2. Set up the synthetic division table: To use synthetic division, we'll use the zero of the divisor. For the divisor [tex]\(x + 5\)[/tex], the zero is [tex]\(-5\)[/tex].
3. Arrange the coefficients: Place the coefficients [tex]\(3, 20, 22, -8\)[/tex] in a row. Write the zero [tex]\(-5\)[/tex] to the left as shown below:
```
-5 | 3 20 22 -8
|
|
```
4. Bring down the leading coefficient: Bring down the first coefficient (3) below the line as it is.
```
-5 | 3 20 22 -8
|
|
3
```
5. Multiply and add: Multiply [tex]\(-5\)[/tex] by the number just brought down (3) and place the result (which is -15) under the next coefficient (20). Then add the column:
```
-5 | 3 20 22 -8
| -15
|
3 5
```
- [tex]\(20 + (-15) = 5\)[/tex]
6. Repeat the process: Multiply [tex]\(-5\)[/tex] by the new result (5), place the result (-25) under the next coefficient (22), and add the column:
```
-5 | 3 20 22 -8
| -15 -25
|
3 5 -3
```
- [tex]\(22 + (-25) = -3\)[/tex]
7. Continue to the last coefficient: Multiply [tex]\(-5\)[/tex] by the latest result [tex]\(-3\)[/tex] and place it under the last coefficient (-8), then add:
```
-5 | 3 20 22 -8
| -15 -25 15
|
3 5 -3 7
```
- [tex]\(-8 + 15 = 7\)[/tex]
8. Interpret the results: The bottom row, excluding the last value, gives the coefficients of the quotient polynomial. The last value is the remainder.
- Quotient: [tex]\( 3x^2 + 5x - 3 \)[/tex]
- Remainder: [tex]\( 7 \)[/tex]
### Conclusion:
The result of the division [tex]\( \frac{3x^3 + 20x^2 + 22x - 8}{x + 5} \)[/tex] is:
[tex]\[ 3x^2 + 5x - 3 \text{ with a remainder of } 7. \][/tex]
In mathematical notation, this can be written as:
[tex]\[ \frac{3x^3 + 20x^2 + 22x - 8}{x + 5} = 3x^2 + 5x - 3 + \frac{7}{x + 5}. \][/tex]
