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nellokov
The question is set up in the following way:

2] 3   0   4   -32

_____________

Here the 2 is from when x-2 is set equal to zero and then solved for x, the rest is from the the number being divided with zero taking the place of the x^2.

Then the following happens:

2] 3   0   4   -32   
         6  12   32
______________
    3   6  16   0

Here the following happens: 
1. We drop down the first coefficient, which is 3
2. We take this coefficient and multiply it by the two in the corner and get 6.
3. We place this six under the second coefficient (0) and add them together and get 6.
4. We take this 6 and multiply it by the two in the corner and get 12.
5. We take this 12 and put it under the third coefficient (4), add them together and get 16.
6. We take the 16 and multiply it by the two in the corner and get 32.
7. We take the 32 and place in under the forth coefficient and get zero.

This gives us the coefficients of the final answer in the order 3, 6 and 16 with no remaninders which gives us 3x^2+6x+16.

Answer:

The quotient of given division is [tex]3x^2+6x+16[/tex].

Step-by-step explanation:

The given expression is

[tex]\frac{3x^3+4x-32}{x-2}[/tex]

The dividend  is

[tex]3x^3+0x^2+4x-32[/tex]

The coefficients of dividend are 3, 0, 4 and -32.

The divisor is (x-2).

If the divisor in the form of (x-c), then we multiply the coefficients and resultant number are multiplied by c. So, the coefficients and resultant number are multiplies by 2.

The steps of synthetic division are:

1. Write the coefficients and 2 in the top row.

2. Write the first coefficient as it is, in bottom row.

3. Multiply the coefficient by 2 and write it in middle row. Now, add them. Apply the same process again.

The bottom row represent the coefficient of quotient and remainder.

The coefficient of quotient are 3,6 and 16. The remainder is 0.

The quotient of given division is [tex]3x^2+6x+16[/tex].

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