Question

find the quotient and remainder using long division.
$\frac{x^3 - 5x - 3}{x - 2}$
the quotient is
the remainder is

Explanation:

Step1: Divide leading terms

$\frac{x^3}{x} = x^2$
Multiply divisor by $x^2$: $x^2(x-2) = x^3 - 2x^2$
Subtract from dividend:
$(x^3 - 5x - 3) - (x^3 - 2x^2) = 2x^2 - 5x - 3$

Step2: Divide new leading terms

$\frac{2x^2}{x} = 2x$
Multiply divisor by $2x$: $2x(x-2) = 2x^2 - 4x$
Subtract from current polynomial:
$(2x^2 - 5x - 3) - (2x^2 - 4x) = -x - 3$

Step3: Divide next leading terms

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

Answer:

The quotient is $x^2 + 2x - 1$
The remainder is $-5$