How would you use synthetic division to find the quotient and remainder for this problem:

(X^3+3ix^2-4ix-2)/(x+i)

To use synthetic division to find the quotient and remainder for the given problem, we will follow these steps:

Step 1: Write down the dividend (the expression being divided) in descending order of exponents. In this case, the dividend is: X^3 + 3iX^2 - 4iX - 2.

Step 2: Identify the divisor. The divisor is x + i.

Step 3: Change the sign of the divisor. Since the divisor is x + i, we will use -(x + i) for synthetic division.

Step 4: Set up the synthetic division table. It should look like this:

-i | 1 3i -4i -2
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1 |
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Step 5: Bring down the first coefficient (1). Write it in the bottom row of the table.

-i | 1 3i -4i -2
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1 | |
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Step 6: Multiply the divisor (-i) by the number in the bottom row (1) and write the result in the next column.

-i | 1 3i -4i -2
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1 | -i |
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Step 7: Add the numbers in the second column. Write the sum in the bottom row of the next column.

-i | 1 3i -4i -2
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1 | -i | 2i |
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Step 8: Repeat steps 6 and 7 until you have filled in all the columns.

-i | 1 3i -4i -2
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1 | -i | 2i | -4 | -2
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Step 9: Now read the bottom row from left to right. The numbers obtained are the coefficients of the quotient. In this case, the quotient is X^2 + 2iX - 4.

Step 10: The last number in the bottom row is the remainder. In this case, the remainder is -2.

Therefore, the quotient is X^2 + 2iX - 4, and the remainder is -2.

The same way you do it normally. Just keep track of the i's and remember that -i^2 = 1

So, the rows of coeffiicients are (munged by font spacing)

1 3i -4i -2 (i
0 i -4 -4i+4
_____________
1 4i -4-4i 2-4i

So that means that

(x^3 + 3ix^2 - 4ix - 2)/(x-i)
= x^2 + 4ix - (4+4i) remainder 2-4i

If you multiply it out, it works.