To divide the polynomial \(12x^2 + 2x - 9\) by \(4x - 2\), we can use polynomial long division.
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Set up the long division: Write \(12x^2 + 2x - 9\) under the long division symbol and \(4x - 2\) outside.
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Determine the first term of the quotient: Divide the leading term \(12x^2\) by the leading term \(4x\): \[ \frac{12x^2}{4x} = 3x \] So, the first term of the quotient is \(3x\).
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Multiply: Multiply \(3x\) by \(4x - 2\): \[ 3x(4x - 2) = 12x^2 - 6x \]
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Subtract: Subtract \(12x^2 - 6x\) from \(12x^2 + 2x - 9\): \[ (12x^2 + 2x - 9) - (12x^2 - 6x) = 2x - 9 + 6x = 8x - 9 \]
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Repeat the process: Now, divide the leading term \(8x\) by the leading term \(4x\): \[ \frac{8x}{4x} = 2 \] So, the next term of the quotient is \(2\).
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Multiply: Multiply \(2\) by \(4x - 2\): \[ 2(4x - 2) = 8x - 4 \]
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Subtract: Subtract \(8x - 4\) from \(8x - 9\): \[ (8x - 9) - (8x - 4) = -9 + 4 = -5 \]
Now, we have a remainder of \(-5\).
Putting it all together, we have: \[ 12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5 \]
Thus, the result of dividing \(12x^2 + 2x - 9\) by \(4x - 2\) is: \[ \text{Quotient: } 3x + 2, \text{ Remainder: } -5 \]