To divide the polynomial \( 4x^3 - 21x^2 + 27x \) by \( x - 3 \) using long division, follow these steps:
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Set Up the Division: Write the dividend \( 4x^3 - 21x^2 + 27x \) and the divisor \( x - 3 \) in long division format.
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Divide the Leading Terms: Divide the leading term of the dividend \( 4x^3 \) by the leading term of the divisor \( x \): \[ \frac{4x^3}{x} = 4x^2 \]
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Multiply and Subtract:
- Multiply \( 4x^2 \) by the entire divisor \( x - 3 \): \[ 4x^2(x - 3) = 4x^3 - 12x^2 \]
- Subtract this 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, divide the leading term of the new polynomial \( -9x^2 \) by \( x \): \[ \frac{-9x^2}{x} = -9x \]
- Multiply \( -9x \) by the divisor \( x - 3 \): \[ -9x(x - 3) = -9x^2 + 27x \]
- Subtract this from the current polynomial: \[ (-9x^2 + 27x) - (-9x^2 + 27x) = 0 \]
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Final Steps: Since there are no remaining terms, the quotient of the division is: \[ \text{Quotient} = 4x^2 - 9x \] and there is no remainder.
Thus, the result of dividing \( 4x^3 - 21x^2 + 27x \) by \( x - 3 \) is: \[ \boxed{4x^2 - 9x} \]
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