To perform the division of the polynomial \(-7x + 6x^3 + 5\) by \(x - 2\), we will use polynomial long division.
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Arranging the polynomials: First, let's write the polynomial in descending order of powers. The dividend is \(6x^3 - 7x + 5\), and the divisor is \(x - 2\).
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Setting up for long division: \[ \begin{array}{r|l} & 6x^3 + 0x^2 - 7x + 5 \ x - 2 & \end{array} \]
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Divide the leading term: We divide the leading term of the dividend \(6x^3\) by the leading term of the divisor \(x\): \[ 6x^3 \div x = 6x^2. \]
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Multiply and subtract: Multiply \(6x^2\) by the entire divisor \(x - 2\) and subtract: \[ (6x^2)(x - 2) = 6x^3 - 12x^2. \] Subtract this from the original polynomial: \[ (6x^3 + 0x^2 - 7x + 5) - (6x^3 - 12x^2) = 0 + 12x^2 - 7x + 5 = 12x^2 - 7x + 5. \]
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Repeat the process: Now, we divide the leading term \(12x^2\) by \(x\): \[ 12x^2 \div x = 12x. \] Multiply \(12x\) by \(x - 2\): \[ (12x)(x - 2) = 12x^2 - 24x. \] Subtract: \[ (12x^2 - 7x + 5) - (12x^2 - 24x) = 0 + 17x + 5 = 17x + 5. \]
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Continue the process: Now divide \(17x\) by \(x\): \[ 17x \div x = 17. \] Multiply \(17\) by \(x - 2\): \[ 17(x - 2) = 17x - 34. \] Subtract: \[ (17x + 5) - (17x - 34) = 0 + 39 = 39. \]
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Final Result: Now we have completed the division. The result can be expressed as: \[ 6x^2 + 12x + 17 + \frac{39}{x - 2}. \]
Therefore, the final answer of the division is: \[ \frac{-7x + 6x^3 + 5}{x - 2} = 6x^2 + 12x + 17 + \frac{39}{x - 2}. \]