To divide the polynomial (7x^3 - 17x^2 - 138x - 72) by (7x + 4), we can use polynomial long division.
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(7x + 4) | 7x^3 - 17x^2 - 138x - 72
Let's start by dividing the first term of the dividend by the first term of the divisor:
7x^3 / 7x = x^2
Now, multiply x^2 by (7x + 4):
x^2 * (7x + 4) = 7x^3 + 4x^2
Subtract this from the original polynomial:
(7x^3 - 17x^2 - 138x - 72) - (7x^3 + 4x^2) = -21x^2 - 138x - 72
Now, bring down the next term:
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(7x + 4) | 7x^3 - 17x^2 - 138x - 72
-21x^2
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-38x - 72
Now, divide the first term of what remains by the first term of the divisor:
-38x / 7x = -5
Multiply -5 by (7x + 4):
-5(7x + 4) = -35x - 20
Subtract this from the current dividend:
(-38x - 72) - (-35x - 20) = -3x - 52
We now bring down the last term:
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(7x + 4) | 7x^3 - 17x^2 - 138x - 72
-21x^2 - 5
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-38x - 72
-38x - 56
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-16
Since the degree of the remainder is less than the degree of the divisor, we have reached the end of the division.
Therefore, the quotient is x^2 - 5 and the remainder is -16.
Thus, (7x^3 - 17x^2 - 138x - 72) / (7x + 4) = x^2 - 5 + (-16)/(7x + 4)
(7x^3 - 17x^2 - 138x - 72) / (7x+4)
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