To divide the polynomial \( 21x^2 - 52x + 32 \) by \( 7x - 8 \), we can use polynomial long division.
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Divide the leading terms: \[ \frac{21x^2}{7x} = 3x \]
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Multiply \( 3x \) by \( 7x - 8 \): \[ 3x(7x - 8) = 21x^2 - 24x \]
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Subtract this from the original polynomial: \[ (21x^2 - 52x + 32) - (21x^2 - 24x) = -52x + 24x + 32 = -28x + 32 \]
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Repeat the process with the new polynomial \( -28x + 32 \):
- Divide the leading term: \[ \frac{-28x}{7x} = -4 \]
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Multiply \( -4 \) by \( 7x - 8 \): \[ -4(7x - 8) = -28x + 32 \]
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Subtract: \[ (-28x + 32) - (-28x + 32) = 0 \]
Since we've reached a remainder of 0, the division is complete.
The quotient when \( 21x^2 - 52x + 32 \) is divided by \( 7x - 8 \) is: \[ \boxed{3x - 4} \]
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