To divide the polynomials \( (8x^4 - 12x^3 + 12x^2 - 4x + 7) \) by \( (4x^2 - 2x) \), we can perform polynomial long division.
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Set Up the Division:
- Dividend: \( 8x^4 - 12x^3 + 12x^2 - 4x + 7 \)
- Divisor: \( 4x^2 - 2x \)
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Divide the Leading Terms:
- Divide \( 8x^4 \) by \( 4x^2 \) to get \( 2x^2 \).
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Multiply the Divisor by the Result:
- \( 2x^2(4x^2 - 2x) = 8x^4 - 4x^3 \).
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Subtract from the Dividend: \[ (8x^4 - 12x^3 + 12x^2 - 4x + 7) - (8x^4 - 4x^3) = (-12x^3 + 4x^3) + 12x^2 - 4x + 7 = -8x^3 + 12x^2 - 4x + 7 \]
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Repeat the Process:
- Now, divide the leading term \( -8x^3 \) by \( 4x^2 \) to get \( -2x \).
- Multiply the divisor by \( -2x \): \[ -2x(4x^2 - 2x) = -8x^3 + 4x^2 \]
- Subtract from the current polynomial: \[ (-8x^3 + 12x^2 - 4x + 7) - (-8x^3 + 4x^2) = (12x^2 - 4x^2) - 4x + 7 = 8x^2 - 4x + 7 \]
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Continue Dividing:
- Divide \( 8x^2 \) by \( 4x^2 \) to get \( 2 \).
- Multiply the divisor by \( 2 \): \[ 2(4x^2 - 2x) = 8x^2 - 4x \]
- Subtract from the current polynomial: \[ (8x^2 - 4x + 7) - (8x^2 - 4x) = 7 \]
Now, we have reached the point where the remainder is less in degree than the divisor.
- Final Result: The division yields: \[ 2x^2 - 2 + \frac{7}{4x^2 - 2x} \]
Thus, the final result of the division \( (8x^4 - 12x^3 + 12x^2 - 4x + 7) \div (4x^2 - 2x) \) is: \[ \boxed{2x^2 - 2 + \frac{7}{4x^2 - 2x}} \]