Explanation:
it follows practically the same rules as a division between plain numbers.
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) = x⁴
we take the first term (x⁵) and divide it by the x term(s) of the divisor (we ignore constants for the moment). the resulting x term : x⁴
as x⁵/x = x⁴
now we multiply the intermediate result with the full divisior and subtract that result from the left side of the dividend :
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) = x⁴
- x⁵ - x⁴
--------------
0 0
now we "pull down" the next term of the dividend and we calculate
(0 + 0 + 9x³) / x = 9x²
so, we have now
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) = x⁴ + 9x²
and we do the same thing as before (multiply intermediate result with divisor and subtract the result)
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) = x⁴ + 9x²
0 0 9x³
- 9x³ - 9x²
---------------
0 0
and, again, we pull down the next term of the dividend abd divide by the x term(s) of the divisor
(0 + 0 + 0 + 0 + 9x) / x = 9
so, we have now
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) = x⁴ + 9x² + 9
and we do the same thing as above again
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) = x⁴ + 9x² + 9
- 0 0 0 0 9x - 9
-----------
0 - 1
since we ran out of additional terms, we are finished.
the quotient is
x⁴ + 9x² + 9
the remainder is
-1
which makes the full division result
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) =
= x⁴ + 9x² + 9 - 1/(x - 1)
let's check :
(x⁴ + 9x² + 9 - 1(x - 1)) × (x - 1) =
= x⁵ + 9x³ + 9x - x⁴ - 9x² - 9 - 1 =
= x⁵ - x⁴ + 9x³ - 9x² + 9x - 10
perfect.