Functions Help?

Question

Here's the problem. I don't get this at all.
Instructions: Express the function in the form
f(x)=(x-k)q(x)+r for the given value of k, and demonstrate that f(k)=r

Function: f(x)=x^3-x^2-14x+11
Value of k: -2

I know how to do synthetic division and come up with x^2-7x+3+(2/x+2) but I dont know what to do now.

Answer 1

If you express the function in the form (x-k)*q(x)+r, it is clear that when x = k, x-k = 0, so the only term left is r. By observation, q(x) will be a polynomial x^2 + ax + b. when this is multiplied by x-k, you will get a polynomial x^3 + ax^2 + bx - kx^2 - akx - bk. This will be your q(x). This + r must equate to the function given, x^3 - x^2 -14x +11

x^3 + (a-k)x^2 + (b-ak)x - bk + r = x^3 - x^2 -14x +11

Equating the coefficients of like powers we get

(a-k) = -1
(b-ak) = -14
(-bk+r) = 11

Your k is given, so you have three unknowns: a, b and r; and three equations. Solve for a, b and r. Then your q(x) and r are known.

EDIT: You get the same answer by dividng f(x) by (x-k); you get the polynomial x^2 + (k-1)x + [k(k-1)-14] with the remainder being r = 11 + k[k(k-1) - 14].