To divide the polynomial \( n^3 + 7n^2 + 14n + 3 \) by \( n + 2 \), we can use polynomial long division.
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Divide the leading term: Divide the leading term of the dividend \( n^3 \) by the leading term of the divisor \( n \): \[ \frac{n^3}{n} = n^2 \]
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Multiply and subtract: Multiply \( n^2 \) by the entire divisor \( n + 2 \) and subtract from the original polynomial: \[ n^3 + 7n^2 + 14n + 3 - (n^2(n + 2)) = n^3 + 7n^2 + 14n + 3 - (n^3 + 2n^2) = 5n^2 + 14n + 3 \]
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Repeat the process: Now take \( 5n^2 \) and divide by \( n \): \[ \frac{5n^2}{n} = 5n \]
Multiply \( 5n \) by \( n + 2 \): \[ 5n(n + 2) = 5n^2 + 10n \] Subtract this from \( 5n^2 + 14n + 3 \): \[ 5n^2 + 14n + 3 - (5n^2 + 10n) = 4n + 3 \]
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Repeat again: Now take \( 4n \) and divide by \( n \): \[ \frac{4n}{n} = 4 \]
Multiply 4 by \( n + 2 \): \[ 4(n + 2) = 4n + 8 \] Subtract this from \( 4n + 3 \): \[ 4n + 3 - (4n + 8) = 3 - 8 = -5 \]
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Conclusion: We have finished dividing. The quotient is: \[ n^2 + 5n + 4 \] and the remainder is: \[ -5 \]
Thus, the answer is: D. The quotient is \( n^2 + 5n + 4 \) and the remainder is -5.