SOLUTION: Use synthetic division to rewrite the following fraction in the form q(x)+r(x)/d(x), where d(x) is the denominator of the original fraction, q(x) is the quotient, and r(x) is the r

Question 1186857: Use synthetic division to rewrite the following fraction in the form q(x)+r(x)/d(x), where d(x) is the denominator of the original fraction, q(x) is the quotient, and r(x) is the remainder.
2x^3 − 8ix^2 + 4x + (9−2i)/x−2i
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here's how to perform synthetic division with complex numbers and rewrite the expression:
**1. Set up the synthetic division:**
Write down the coefficients of the polynomial in the numerator and the value of x that makes the denominator zero (2i in this case):
```
2i | 2 -8i 4 9 - 2i
|________________________
```
**2. Perform the synthetic division:**
* Bring down the first coefficient (2):
```
2i | 2 -8i 4 9 - 2i
|
------------------------
2
```
* Multiply the value outside (2i) by the number just brought down (2) and write the result under the next coefficient (-8i):
```
2i | 2 -8i 4 9 - 2i
| 4i
------------------------
2 -4i
```
* Add the numbers in the second column (-8i and 4i):
```
2i | 2 -8i 4 9 - 2i
| 4i
------------------------
2 -4i
```
* Multiply 2i by -4i and write the result under the next coefficient (4):
```
2i | 2 -8i 4 9 - 2i
| 4i 8
------------------------
2 -4i 12
```
* Add the numbers in the third column (4 and 8):
```
2i | 2 -8i 4 9 - 2i
| 4i 8
------------------------
2 -4i 12
```
* Multiply 2i by 12 and write the result under the last term (9-2i):
```
2i | 2 -8i 4 9 - 2i
| 4i 8 24i
------------------------
2 -4i 12 9 + 22i
```
* Add the numbers in the last column:
```
2i | 2 -8i 4 9 - 2i
| 4i 8 24i
------------------------
2 -4i 12 9 + 22i
```
**3. Interpret the results:**
The numbers in the bottom row (2, -4i, 12) are the coefficients of the quotient, and the last number (9 + 22i) is the remainder. The degree of the quotient is one less than the degree of the original polynomial.
Therefore, the quotient q(x) is 2x² - 4ix + 12, and the remainder r(x) is 9 + 22i. The divisor d(x) is x - 2i.
**4. Write the expression:**
The expression can be written as:
2x² - 4ix + 12 + (9 + 22i) / (x - 2i)

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