can somebody please clearly show me how to solve these problems?

Question

they're polynomial division problems:

x^3 + 2x^2 + 3x - 6 divided by x - 1


and


n^3 - 27 / n - 3

Answer 1

EDITED:

You do them by synthetic division which is like arithmetic long division
......-------------------------
x-1 | x^3 +2x^2 + 3x - 6

Divide the first term of the dividend polynomial by the first term of the divisor polynomial to get the first term of the quotient. The first term of the quotient is x^2, since (x^3)/x = x^2:

........x^2
......-------------------------
x-1 | x^3 +2x^2 + 3x - 6

Then multiply as in long division:

........x^2
......-------------------------
x-1 | x^3 +2x^2 + 3x - 6
........x^3 - x^2

and subtract the product from the dividend polynomial

........x^2
......-------------------------
x-1 | x^3 +2x^2 + 3x - 6
........x^3 - x^2
........-------------------------
................3x^2 + 3x - 6.. ............(intermediate polynomial)

and do it again using the intermediate polynomial as a dividend:

.......x^2 + 3x + 6
......-----------------------
x-1 | x^3 +2x^2 + 3x - 6
........x^3 - x^2
.......------------------------
................3x^2 + 3x - 6
................3x^2 - 3x
................-----------------
...........................6x - 6
...........................6x - 6
..........................-----------
............................0.....0


The second problem is done the same way--just put in zeros for the missing powers in the polynomial

.....n^2 + 3n + 9
.....-------------------------------
n-3|n^3 + 0[n^2] + 0[n] - 27
......n^3..- 3n^2
.....--------------------------------
...............3n^2 + 0[n] - 27
................3n^2 - 9n
...............------------------------
...........................9n - 27
...........................9n - 27
..........................----------
............................0.....0