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
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
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!
Good job with the formatting!!
Did you post somewhere in the past as a scratchpad?
To write the function f(x) = x^3 - 11x^2 + 18x + 32 in the form f(x) = (x - k)q(x) + r, where k = 6 + √6, we'll need to perform polynomial long division and evaluate f(k) to find the remainder r.
First, let's find the quotient q(x) using polynomial long division. We divide the polynomial f(x) by (x - k) repeatedly to find q(x) and the remainder r:
(x^2 - 6x - 4)
_______________________
(x - (6 + √6)) | x^3 - 11x^2 + 18x + 32
Start by dividing x into x^3 to get x^2. Then multiply (x - k) by x^2 to get x^3 - kx^2. Subtract this from the original polynomial to get the remainder:
(x^2 - 6x - 4)
_______________________
(x - (6 + √6)) | x^3 - 11x^2 + 18x + 32
- ( x^3 - 6x^2 - √6x^2)
Simplifying the remainder gives us:
(x^2 - 6x - 4 + √6x^2)
_______________________
(x - (6 + √6)) | x^3 - 11x^2 + 18x + 32
To evaluate f(k), substitute x = k into the original polynomial f(x) and simplify:
f(k) = (k)^3 - 11(k)^2 + 18(k) + 32
= (6 + √6)^3 - 11(6 + √6)^2 + 18(6 + √6) + 32
Evaluated f(k), we'll have a value for comparision purposes against the found remainder:
Now, calculate f(k) using the given value of k:
f(k) = (6 + √6)^3 - 11(6 + √6)^2 + 18(6 + √6) + 32
Upon simplification, the value of f(k) is obtained.