To divide the polynomial \(4x^3 - 21x^2 + 27x\) by \(x - 3\) using long division, we will perform the following steps:
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Set up the division: Write the dividend \(4x^3 - 21x^2 + 27x\) under the long division symbol, and the divisor \(x - 3\) outside.
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Divide the leading term: Divide the leading term of the dividend \(4x^3\) by the leading term of the divisor \(x\) to get \(4x^2\).
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Multiply and subtract: Multiply \(4x^2\) by the entire divisor \(x - 3\): \[ 4x^2 \cdot (x - 3) = 4x^3 - 12x^2 \] Now, subtract this result from the original polynomial: \[ (4x^3 - 21x^2 + 27x) - (4x^3 - 12x^2) = -21x^2 + 12x^2 + 27x = -9x^2 + 27x \]
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Repeat the process: Now repeat the division with the new polynomial \(-9x^2 + 27x\). Divide \(-9x^2\) by \(x\): \[ -9x^2 \div x = -9x \] Multiply \(-9x\) by the divisor: \[ -9x \cdot (x - 3) = -9x^2 + 27x \] Subtract this from \(-9x^2 + 27x\): \[ (-9x^2 + 27x) - (-9x^2 + 27x) = 0 \]
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Conclusion: At this point, we have no remainder left to divide by the divisor \(x - 3\). Therefore, we have completed the long division.
The final result from dividing the polynomials \(4x^3 - 21x^2 + 27x\) by \(x - 3\) is:
\[ \boxed{4x^2 - 9x} \]
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