Question 1-13 Divide (x^4-7x^3+7x^2-2x+6)/(x-1) x^3-6x^2+x+4 x^3+6x^2+x-1+(5)/(x-1) x^3-8x^2+x-1+(4)/(x-1) x^3-6x^2+x-1+(5)/(x-1)

To divide the polynomial $\frac{x^{4}-7x^{3}+7x^{2}-2x+6}{x-1}$, we can use polynomial long division.<br /><br />Step 1: Divide the first term of the numerator by the denominator.<br />$\frac{x^{4}}{x-1} = x^{3}$<br /><br />Step 2: Multiply the denominator by the result from Step 1 and subtract it from the numerator.<br />$(x-1) \cdot x^{3} = x^{4}-x^{3}$<br />$(x^{4}-7x^{3}+7x^{2}-2x+6) - (x^{4}-x^{3}) = -6x^{3}+7x^{2}-2x+6$<br /><br />Step 3: Repeat Step 1 with the new numerator.<br />$\frac{-6x^{3}}{x-1} = -6x^{2}$<br /><br />Step 4: Multiply the denominator by the result from Step 3 and subtract it from the new numerator.<br />$(x-1) \cdot -6x^{2} = -6x^{3}+6x^{2}$<br />$(-6x^{3}+7x^{2}-2x+6) - (-6x^{3}+6x^{2}) = x^{2}-2x+6$<br /><br />Step 5: Repeat Step 1 with the new numerator.<br />$\frac{x^{2}}{x-1} = x$<br /><br />Step 6: Multiply the denominator by the result from Step 5 and subtract it from the new numerator.<br />$(x-1) \cdot x = x^{2}-x$<br />$(x^{2}-2x+6) - (x^{2}-x) = -x+6$<br /><br />Step 7: Repeat Step 1 with the new numerator.<br />$\frac{-x}{x-1} = -1$<br /><br />Step 8: Multiply the denominator by the result from Step 7 and subtract it from the new numerator.<br />$(x-1) \cdot -1 = -x+1$<br />$(-x+6) - (-x+1) = 5$<br /><br />Therefore, the result of the division is:<br />$x^{3}-6x^{2}+x+4+\frac{5}{x-1}$<br /><br />So, the correct answer is:<br />$x^{3}-6x^{2}+x+4+\frac{5}{x-1}$