Write f(x) =x^3-11x^2+18x+32 in the form f(x)= (x-k) q (x) +r when k=6 + (square root)6, and demonstrate that f(k)=r

Answer 1

A little synthetic division shows that

f(x) = (x-(6+√6))(x^2+(√6-5)x+(√6-6))+2

I leave it to you to verify that
f(6+√6) = 2

Answer 2

I will try to show my synthetic division, the .... are spaces:

6+√6 | 1 ........ -11 ........ 18 ........ 32
.................... 6+√6 .... √6-24 .... -30
............ 1 ..... √6-5 ..... √6-6 ........ 2

showing that:
x^3-11x^2+18x+32 = (x-6-√6)(x^2 + (√6-5)x + √6-6) + 2

that is, r = 2

now find f(6+√6)
= (6+√6)^3 - 11(√6+6)^2 + 18(√6+6) + 32
= 324+114√6 - 462 - 132√6 + 18√6 + 108 + 32
= 2 + 0√6
= 2

whewwwhh!

Answer 3

Good job with the formatting!!

Did you post somewhere in the past as a scratchpad?