To divide the polynomial \( 8x^4 - 12x^3 + 12x^2 - 4x + 7 \) by \( 4x^2 - 2x \), we can use polynomial long division. Here’s how the division proceeds:
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Divide the leading terms: Divide the leading term of the dividend (\( 8x^4 \)) by the leading term of the divisor (\( 4x^2 \)): \[ \frac{8x^4}{4x^2} = 2x^2 \]
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Multiply: Multiply \( 2x^2 \) by the entire divisor \( 4x^2 - 2x \): \[ 2x^2(4x^2 - 2x) = 8x^4 - 4x^3 \]
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Subtract: Subtract this result from the original polynomial: \[ (8x^4 - 12x^3 + 12x^2 - 4x + 7) - (8x^4 - 4x^3) = (-12x^3 + 4x^3 + 12x^2 - 4x + 7) = -8x^3 + 12x^2 - 4x + 7 \]
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Repeat: Now repeat the process with the new polynomial \(-8x^3 + 12x^2 - 4x + 7\).
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Divide the leading terms: \[ \frac{-8x^3}{4x^2} = -2x \]
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Multiply: \[ -2x(4x^2 - 2x) = -8x^3 + 4x^2 \]
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Subtract: \[ (-8x^3 + 12x^2 - 4x + 7) - (-8x^3 + 4x^2) = 12x^2 - 4x^2 - 4x + 7 = 8x^2 - 4x + 7 \]
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Continue: Continue with \( 8x^2 - 4x + 7 \).
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Divide the leading terms: \[ \frac{8x^2}{4x^2} = 2 \]
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Multiply: \[ 2(4x^2 - 2x) = 8x^2 - 4x \]
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Subtract: \[ (8x^2 - 4x + 7) - (8x^2 - 4x) = 7 \]
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Now, we have reached a remainder of \( 7 \).
Final Result
The result of the division is: \[ 2x^2 - 2x + 2 \quad \text{with a remainder of } 7 \] Thus, the final answer is: \[ \frac{8x^4 - 12x^3 + 12x^2 - 4x + 7}{4x^2 - 2x} = 2x^2 - 2x + 2 + \frac{7}{4x^2 - 2x} \]