To find the quotient of the polynomials \(-6x^3 + 17x^2 - 32x + 9\) divided by \(2x^2 - 5x + 9\) using long division, we can follow these steps:
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Arrange the polynomials in standard form: The dividend (the polynomial being divided) is \[ -6x^3 + 17x^2 - 32x + 9 \] and the divisor (the polynomial to divide by) is \[ 2x^2 - 5x + 9 \]
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Perform polynomial long division:
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Divide the leading term of the dividend \(-6x^3\) by the leading term of the divisor \(2x^2\): \[ \frac{-6x^3}{2x^2} = -3x \]
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Multiply the entire divisor \(2x^2 - 5x + 9\) by \(-3x\): \[ -3x(2x^2 - 5x + 9) = -6x^3 + 15x^2 - 27x \]
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Subtract this from the original dividend: \[ (-6x^3 + 17x^2 - 32x + 9) - (-6x^3 + 15x^2 - 27x) = (17x^2 - 15x^2) + (-32x + 27x) + 9 \] \[ = 2x^2 - 5x + 9 \]
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Repeat the process:
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Now divide the leading term \(2x^2\) by the leading term \(2x^2\): \[ \frac{2x^2}{2x^2} = 1 \]
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Multiply the entire divisor by \(1\): \[ 1(2x^2 - 5x + 9) = 2x^2 - 5x + 9 \]
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Subtract this from the previous remainder: \[ (2x^2 - 5x + 9) - (2x^2 - 5x + 9) = 0 \]
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Since the remainder is \(0\), the process stops here.
- Write the result: The quotient is \(-3x + 1\).
Thus, the coefficients of the quotient \(-3x + 1\) are:
- For \(x^1\): \(-3\)
- For \(x^0\): \(1\)
Finally, the answer is: \[ \boxed{[-3, 1]} \]