Polynomial Question?

The cubic polynomial f(x) is such that the coefficient of x^3 is -1 and the roots of the equation f(x)=0 are 1, 2 and k. Given that f(x) has a remainder of 8 when divided by x-3, find:
i. The value of k
ii. The remainder when f(x) is divided by x+3

1 Answer
Aug 4, 2018

k=7 and the remainder when divided by x+3 is 200

Explanation:

Given that f(x) has zeros 1, 2, k, it has factors (x-1), (x-2) and (x-k).

So with the additional information that the coefficient of the leading term is -1, we can write:

f(x) = -(x-1)(x-2)(x-k)

The remainder 8 when divided by (x-3) is the value of f(3), so:

8 = f(3)

color(white)(8) = -((color(blue)(3))-1)((color(blue)(3))-2)((color(blue)(3))-k)

color(white)(8) = 2k-6

Hence k=(8+6)/2 = 7

Then the remainder when divided by (x+3) is f(-3):

f(-3) = -((color(blue)(-3))-1)((color(blue)(-3))-2)((color(blue)(-3))-7)

color(white)(f(-3)) = 200

graph{(y+(x-1)(x-2)(x-7))=0 [-4, 10, -15, 32]}