To perform the long division of the polynomials \( 4x^3 - 21x^2 + 27x - 3 \) by \( 4x^2 - 33x + 126 \), we will follow the steps of polynomial long division.
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Set up the division: Write \( 4x^3 - 21x^2 + 27x - 3 \) under the long division symbol and \( 4x^2 - 33x + 126 \) outside.
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Divide the first term: Divide the leading term of the dividend \( 4x^3 \) by the leading term of the divisor \( 4x^2 \): \[ \frac{4x^3}{4x^2} = x \] Write \( x \) above the division line.
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Multiply: Multiply \( x \) by the entire divisor: \[ x(4x^2 - 33x + 126) = 4x^3 - 33x^2 + 126x \]
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Subtract: Subtract this result from the original polynomial: \[ (4x^3 - 21x^2 + 27x - 3) - (4x^3 - 33x^2 + 126x) = (12x^2 - 99x - 3) \]
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Repeat: Now, divide the leading term \( 12x^2 \) by the leading term of the divisor \( 4x^2 \): \[ \frac{12x^2}{4x^2} = 3 \] Write \( +3 \) above the division line next to \( x \).
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Multiply and subtract again: Multiply \( 3 \) by the entire divisor: \[ 3(4x^2 - 33x + 126) = 12x^2 - 99x + 378 \] Now subtract this from the previous remainder: \[ (12x^2 - 99x - 3) - (12x^2 - 99x + 378) = -381 \]
After this step, we can summarize the division process so far:
Quotient: \( x + 3 \)
Remainder: \( -381 \)
Finally, we can express the result of the polynomial division as: \[ \frac{4x^3 - 21x^2 + 27x - 3}{4x^2 - 33x + 126} = x + 3 - \frac{381}{4x^2 - 33x + 126} \]
And that is the complete solution to the polynomial long division.
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