Bot GPT 4-o mini To divide the polynomial \(4x^4 + 4x - 10\) by \(2x^2 - 3\), we can use polynomial long division.
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Set up the division: \[ \frac{4x^4 + 0x^3 + 0x^2 + 4x - 10}{2x^2 - 3} \]
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Divide the leading terms: Divide \(4x^4\) (the leading term of the dividend) by \(2x^2\) (the leading term of the divisor): \[ \frac{4x^4}{2x^2} = 2x^2 \]
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Multiply and subtract: Multiply \(2x^2\) by the entire divisor \(2x^2 - 3\): \[ 2x^2(2x^2 - 3) = 4x^4 - 6x^2 \] Now subtract this from the original polynomial: \[ (4x^4 + 0x^3 + 0x^2 + 4x - 10) - (4x^4 - 6x^2) = 0x^4 + 0x^3 + 6x^2 + 4x - 10 \] This simplifies to: \[ 6x^2 + 4x - 10 \]
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Divide again: Now divide the leading term \(6x^2\) by \(2x^2\): \[ \frac{6x^2}{2x^2} = 3 \]
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Multiply and subtract: Multiply \(3\) by the entire divisor \(2x^2 - 3\): \[ 3(2x^2 - 3) = 6x^2 - 9 \] Now subtract: \[ (6x^2 + 4x - 10) - (6x^2 - 9) = 0x^2 + 4x - 1 \] This simplifies to: \[ 4x - 1 \]
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Conclusion: Since the degree of \(4x - 1\) (which is 1) is less than the degree of the divisor \(2x^2 - 3\) (which is 2), we stop here.
Thus, the quotient is \(2x^2 + 3\) and the remainder is \(4x - 1\).
Therefore, the correct response is: The quotient is \(2x^2 + 3\), the remainder is \(4x - 1\).
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