Using synthetic division:
x^3 + 4x^2 - 3x - 12 / x^2 - 3
Write it out like a long division problem. x^-3 goes into x^3 + 4x^2 -3x -12x times. This is the first term of the answer. Multiply x by x^2 -3 and write the product under the dividend. That would be x^3 -3x. Subtract x^3 - 3x from the dividend and you have 4x^2 -12 left over. How many times does x-3 go into that? Add that to the x in the quotient.
The answer is x+4
Well, well, well, looks like we've got quite the division problem here! Let's gather our wits and get into action.
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x^2 - 3 | x^3 + 4x^2 - 3x - 12
Alright, folks, let's start by figuring out how many times x^2 - 3 goes into x^3 + 4x^2 - 3x - 12. We can simply take a wild guess and say it goes in there x times. Good start, right?
Now, let's multiply x by x^2 - 3 and write the product under the dividend:
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x^2 - 3 | x^3 + 4x^2 - 3x - 12
- (x^3 - 3x)
Oooh, subtraction time! Let's subtract the result (x^3 - 3x) from the dividend (x^3 + 4x^2 - 3x - 12):
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x^2 - 3 | x^3 + 4x^2 - 3x - 12
- (x^3 - 3x)
_____________________
4x^2 - 12
Now, we have 4x^2 - 12 left. What a twist! Let's determine how many times x^2 - 3 fits into that remainder. Like Sherlock Holmes, we'll let our intuition guide us. It turns out that x = 4 is the answer!
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x^2 - 3 | x^3 + 4x^2 - 3x - 12
- (x^3 - 3x)
_____________________
4x^2 - 12
- ( 4x^2 - 12)
______________________
0
Hooray! The remainder is zero, so we can confidently say that the answer is...
*drumroll please*
x + 4! Ta-da!
Hope that brings a smile to your face. Divisions can be quite the rollercoaster ride sometimes, but we made it through with our clownish antics intact!
To perform synthetic division, we can rewrite the division problem as:
(x^3 + 4x^2 - 3x - 12) ÷ (x^2 - 3)
Now we can set up synthetic division:
x + 4
x^2 - 3 | x^3 + 4x^2 - 3x - 12
Let's start by writing down the coefficients of the dividend:
x + 4
x^2 - 3 | 1 4 -3 -12
We need to bring down the first coefficient, which is 1:
x + 4
x^2 - 3 | 1 4 -3 -12
1
Next, we multiply the divisor (x^2 - 3) by the first term of the quotient (x), and write the result below the line:
x + 4
x^2 - 3 | 1 4 -3 -12
1 1
Now, we add the second column of numbers:
x + 4
x^2 - 3 | 1 4 -3 -12
1 1
_____________
1 5
Next, we repeat the process by multiplying the divisor (x^2 - 3) by the new term in the quotient (5), and write the result below the line:
x + 4
x^2 - 3 | 1 4 -3 -12
1 1
_____________
1 5
5 20
We then add the third column of numbers:
x + 4
x^2 - 3 | 1 4 -3 -12
1 1
_____________
1 5
5 20
_____________
1 5 17
Now we have obtained the remainder of 17, and since we have reached the second degree (x^2), we stop the process. The quotient is represented by the terms above the line, which are x + 4. Therefore, the answer is x + 4.
Note: It seems there was an error in your question where you mentioned x^-3. However, in the provided equation, the term x^-3 does not appear.
To solve the division problem using synthetic division, you need to follow these steps:
1. Write the dividend (x^3 + 4x^2 - 3x - 12) and the divisor (x^2 - 3) in the proper format for long division.
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x^2 - 3 | x^3 + 4x^2 - 3x - 12
2. Determine the first term of the quotient by dividing the leading term of the dividend (x^3) by the leading term of the divisor (x^2). In this case, x^3 ÷ x^2 = x.
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x^2 - 3 | x^3 + 4x^2 - 3x - 12
x
3. Multiply the divisor (x^2 - 3) by the first term of the quotient (x), and write the product below the dividend.
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x^2 - 3 | x^3 + 4x^2 - 3x - 12
- (x^3 - 3x)
7x^2 - 3x
4. Subtract the product from the dividend. In this case, subtract (x^3 - 3x) from (x^3 + 4x^2 - 3x -12), which results in 7x^2 - 3x.
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x^2 - 3 | x^3 + 4x^2 - 3x - 12
- (x^3 - 3x)
__________________
7x^2 - 3x
5. Repeat steps 2-4 with the remainder obtained. Now, divide the leading term of the new dividend (7x^2) by the leading term of the divisor (x^2), which gives 7x^2 ÷ x^2 = 7.
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x^2 - 3 | x^3 + 4x^2 - 3x - 12
- (x^3 - 3x)
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7x^2 - 3x
- (7x^2 - 21)
18x - 12
6. Finally, repeat steps 2-4 with the new remainder (18x - 12). Since the divisor (x^2 - 3) is of degree 2 and the remainder is linear (degree 1), the division stops at this point.
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x^2 - 3 | x^3 + 4x^2 - 3x - 12
- (x^3 - 3x)
__________________
7x^2 - 3x
- (7x^2 - 21)
__________________
18x - 12
Therefore, the answer to the division problem is x + 4, where 'x' represents the quotient and "+ 4" represents the constant term.