To simplify the expression \( \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3 - x^2 + 4x} \), first, let's simplify the denominator.
First, combine like terms in the denominator: \[ 2x - 3 - x^2 + 4x = 6x - x^2 - 3 = -x^2 + 6x - 3 \] Thus, the expression can be rewritten as: \[ \frac{8x^3 - 12x^2 + 18x - 27}{-x^2 + 6x - 3} \]
Now to simplify this, we need to perform polynomial long division or factor if possible.
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Factor the denominator \(-x^2 + 6x - 3\): We can factor this by finding the roots using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-6 \pm \sqrt{6^2 - 4 \cdot (-1) \cdot (-3)}}{2 \cdot (-1)} = \frac{-6 \pm \sqrt{36 - 12}}{-2} = \frac{-6 \pm \sqrt{24}}{-2} \] \[ = \frac{-6 \pm 2\sqrt{6}}{-2} = 3 \mp \sqrt{6} \]
Thus, the roots are \(3 - \sqrt{6}\) and \(3 + \sqrt{6}\). The factorization of the quadratic is: \[ -1 \cdot (x - (3 - \sqrt{6}))(x - (3 + \sqrt{6})) \] or simply: \[ -(x^2 - 6x + 3) \]
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Perform polynomial long division for the numerator \(8x^3 - 12x^2 + 18x - 27\) with the factored form of the denominator (or just the quadratic, \( -x^2 + 6x - 3\)): We'll divide the leading term: \[ \frac{8x^3}{-x^2} = -8x \]
Multiply \( -8x \) by the entire denominator: \[ -8x(-x^2 + 6x - 3) = 8x^3 - 48x^2 + 24x \]
Subtract this from the numerator: \[ (8x^3 - 12x^2 + 18x - 27) - (8x^3 - 48x^2 + 24x) = ( -12x^2 + 48x^2) + (18x - 24x) - 27 = 36x^2 - 6x - 27 \]
Now we divide \( 36x^2 - 6x - 27 \) by \( -x^2 + 6x - 3 \).
Next, repeat the process: Divide \( 36x^2 \) by \( -x^2 \) to get \( -36 \). \[ -36(-x^2 + 6x - 3) = 36x^2 - 216x + 108 \] Subtract this: \[ (36x^2 - 6x - 27) - (36x^2 - 216x + 108) = - 6x + 216x - 27 - 108 = 210x - 135 \]
Nice, our remainder is \( 210x - 135 \).
Thus, putting it all together: \[ \frac{8x^3 - 12x^2 + 18x - 27}{-(x^2 - 6x + 3)} = -8x - 36 + \frac{210x - 135}{-(x^2 - 6x + 3)} \] We can leave it in this form or further factor the remainder \( 210x - 135 \).
Final result (depending on how simplified you want it): \[ -8x - 36 + \frac{135 - 210x}{x^2 - 6x + 3} \]
This is the simplified or factored form depending on further instructions required.
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