Question

divide.
\\(\frac{2x^{4} - 16x^{3} + 25x^{2} - 7x + 6}{x - 6}\\)
\\(\frac{2x^{4} - 16x^{3} + 25x^{2} - 7x + 6}{x - 6} = \square\\)

Explanation:

Step1: Use Polynomial Long Division

Divide the leading term of the numerator \(2x^4\) by the leading term of the denominator \(x\) to get \(2x^3\). Multiply the denominator \(x - 6\) by \(2x^3\) to get \(2x^4 - 12x^3\). Subtract this from the numerator: \((2x^4 - 16x^3 + 25x^2 - 7x + 6) - (2x^4 - 12x^3) = -4x^3 + 25x^2 - 7x + 6\).

Step2: Continue the Division

Divide the leading term of the new numerator \(-4x^3\) by \(x\) to get \(-4x^2\). Multiply \(x - 6\) by \(-4x^2\) to get \(-4x^3 + 24x^2\). Subtract: \((-4x^3 + 25x^2 - 7x + 6) - (-4x^3 + 24x^2) = x^2 - 7x + 6\).

Step3: Continue the Division

Divide the leading term \(x^2\) by \(x\) to get \(x\). Multiply \(x - 6\) by \(x\) to get \(x^2 - 6x\). Subtract: \((x^2 - 7x + 6) - (x^2 - 6x) = -x + 6\).

Step4: Final Division Step

Divide the leading term \(-x\) by \(x\) to get \(-1\). Multiply \(x - 6\) by \(-1\) to get \(-x + 6\). Subtract: \((-x + 6) - (-x + 6) = 0\).

Combining the terms from each step: \(2x^3 - 4x^2 + x - 1\).

Answer:

\(2x^3 - 4x^2 + x - 1\)