To perform long division on the expression \( x^3 + 4x^2 + 7x - 5 \) by \( x + 3 \), we will follow the steps for polynomial long division.
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Divide the leading term: Divide the leading term of the dividend, \( x^3 \), by the leading term of the divisor, \( x \): \[ x^3 \div x = x^2 \]
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Multiply and subtract: Multiply \( x^2 \) by the entire divisor \( x + 3 \) and subtract it from the dividend. \[ (x^3 + 4x^2 + 7x - 5) - (x^2(x + 3)) = (x^3 + 4x^2 + 7x - 5) - (x^3 + 3x^2) \] This simplifies to: \[ (4x^2 - 3x^2) + 7x - 5 = x^2 + 7x - 5 \]
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Repeat: Now, repeat the process with \( x^2 + 7x - 5 \).
- Divide \( x^2 \) by \( x \): \[ x^2 \div x = x \]
- Multiply and subtract: \[ (x^2 + 7x - 5) - (x(x + 3)) = (x^2 + 7x - 5) - (x^2 + 3x) \] This simplifies to: \[ (7x - 3x) - 5 = 4x - 5 \]
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Continue to the next term: Now, repeat with \( 4x - 5 \).
- Divide \( 4x \) by \( x \): \[ 4x \div x = 4 \]
- Multiply and subtract: \[ (4x - 5) - (4(x + 3)) = (4x - 5) - (4x + 12) \] This simplifies to: \[ -5 - 12 = -17 \]
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Final result: The quotient is \( x^2 + x + 4 \) and the remainder is \( -17 \). So, we can write this as: \[ \frac{x^3 + 4x^2 + 7x - 5}{x + 3} = x^2 + x + 4 - \frac{17}{x + 3} \]
Hence, the correct response is: \( x^2 + x + 4 - \frac{17}{x + 3} \).