Question

1. find the quotient using long division. express the answer as a rational expression.

overline{x - 1\big)x^3 + x^2 + 3x + 10}
options:
$x^2 + 2x + 20(x - 1)$
$x^2 + 2x + 5 + \frac{15}{x}$
$x^2 + 2x + 5 + 15$
$x + 2x + 5 + \frac{15}{x - 1}$

Explanation:

Step1: Divide leading terms

$\frac{x^3}{x} = x^2$
Multiply divisor by $x^2$: $x^2(x-1) = x^3 - x^2$
Subtract from dividend:
$(x^3 + x^2 + 3x + 10) - (x^3 - x^2) = 2x^2 + 3x + 10$

Step2: Divide new leading terms

$\frac{2x^2}{x} = 2x$
Multiply divisor by $2x$: $2x(x-1) = 2x^2 - 2x$
Subtract from current polynomial:
$(2x^2 + 3x + 10) - (2x^2 - 2x) = 5x + 10$

Step3: Divide new leading terms

$\frac{5x}{x} = 5$
Multiply divisor by $5$: $5(x-1) = 5x - 5$
Subtract from current polynomial:
$(5x + 10) - (5x - 5) = 15$

Step4: Write final quotient

Combine terms, add remainder over divisor.
Quotient: $x^2 + 2x + 5 + \frac{15}{x-1}$

Answer:

$\boldsymbol{x^2 + 2x + 5 + \frac{15}{x-1}}$ (matches the fourth option)