What is the value of n in the expession x cubed plus x squared minus 2nx plus n squared which has a remainder of 8 when divided by x minus 2?

Answer 1

n is 2.

To solve, do the division:

. . . . . . . . .x2 +. 3x + (6-2n)

. . . --------------------------

x-2 | x3 + x2 - 2nx + n2

. . . . .x3 -2x2

. . . . .--------

. . . . . . . .3x2 - 2nx

. . . . . . . .3x2 - . 6x

. . . . . . . .----------

. . . . . . . . . (6-2n)x + n2

. . . . . . . . . (6-2n)x - 2(6-2n)

. . . . . . . . . ----------------------

. . . . . . . . . . . . . . . .n2 + 2(6-2n)

But this remainder is known to be 8, so:

n2 + 2(6-2n) = 8

⇒ n2 - 4n + 4 = 0

⇒ (n - 2)2 = 0

⇒ n = 2