Simplify (x^3-6)/(x-1) using long division. Select all correct ways of expressing this quotient. x^2+x+1+(-5)/(x-1) x^2+x+1+(5)/(x-1) x^2+x+1+(5)/(1-x) D x^2+x+1-(5)/(x-1) x^2+x+1-(5)/(1-x)

To simplify $\frac{x^{3}-6}{x-1}$ using long division, we divide $x^{3}-6$ by $x-1$.<br /><br />Step 1: Divide the first term of the numerator by the first term of the denominator: $\frac{x^{3}}{x} = x^{2}$.<br /><br />Step 2: Multiply the entire denominator by the result from Step 1: $x^{2}(x-1) = x^{3}-x^{2}$.<br /><br />Step 3: Subtract the result from Step 2 from the numerator: $(x^{3}-6) - (x^{3}-x^{2}) = x^{2}-6$.<br /><br />Step 4: Divide the first term of the new numerator by the first term of the denominator: $\frac{x^{2}}{x} = x$.<br /><br />Step 5: Multiply the entire denominator by the result from Step 4: $x(x-1) = x^{2}-x$.<br /><br />Step 6: Subtract the result from Step 5 from the new numerator: $(x^{2}-6) - (x^{2}-x) = x-6$.<br /><br />Step 7: Divide the first term of the new numerator by the first term of the denominator: $\frac{x}{x} = 1$.<br /><br />Step 8: Multiply the entire denominator by the result from Step 7: $1(x-1) = x-1$.<br /><br />Step 9: Subtract the result from Step 8 from the new numerator: $(x-6) - (x-1) = -5$.<br /><br />Step 10: Write the final answer as the quotient plus the remainder divided by the denominator: $x^{2}+x+1+\frac{-5}{x-1}$.<br /><br />Therefore, the correct ways of expressing this quotient are:<br />- $x^{2}+x+1+\frac{-5}{x-1}$<br />- $x^{2}+x+1-\frac{5}{1-x}$