Answer:
The quotient is x^4 - 2x^3 - 9x^2 - 3x with a remainder of -34.
Explanation:
To divide the polynomial x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34 by x + 8 using long division, we can follow these steps:
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x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34
Step 1: Divide the first term of the dividend (x^5) by the first term of the divisor (x), which gives x^4. Write this as the first term of the quotient above the line.
x^4
x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34
Step 2: Multiply the divisor (x + 8) by the quotient term (x^4). Write the result below the dividend, and subtract it from the dividend.
x^4
x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34
- (x^5 + 8x^4)
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7x^4 + 54x^3 - 25x^2 - 75x
Step 3: Bring down the next term of the dividend, which is 54x^3. Now we have a new dividend.
x^4
x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34
- (x^5 + 8x^4)
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7x^4 + 54x^3 - 25x^2 - 75x
- (7x^4 + 56x^3)
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-2x^3 - 25x^2 - 75x
Step 4: Divide the new first term of the dividend (-2x^3) by the first term of the divisor (x), which gives -2x^2. Write this as the next term of the quotient above the line.
x^4 - 2x^3
x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34
- (x^5 + 8x^4)
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7x^4 + 54x^3 - 25x^2 - 75x
- (7x^4 + 56x^3)
_______________________
-2x^3 - 25x^2 - 75x
Step 5: Multiply the divisor (x + 8) by the new quotient term (-2x^3). Write the result below the previous difference, and subtract it from the previous difference.
x^4 - 2x^3
x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34
- (x^5 + 8x^4)
_______________________
7x^4 + 54x^3 - 25x^2 - 75x
- (7x^4 + 56x^3)
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-2x^3 - 25x^2 - 75x
- (-2x^3 - 16x^2)
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-9x^2 - 75x
Step 6: Bring down the next term of the dividend, which is -34. Now we have a new dividend.
x^4 - 2x^3
x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34
- (x^5 + 8x^4)
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7x^4 + 54x^3 - 25x^2 - 75x
- (7x^4 + 56x^3)
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-2x^3 - 25x^2 - 75x
- (-2x^3 - 16x^2)
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-9x^2 - 75x
- (-9x^2 - 72x)
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-3x - 34
Step 7: The division is complete. The final result is -3x - 34.
Therefore, the quotient is x^4 - 2x^3 - 9x^2 - 3x with a remainder of -34.