Meadow,
This is an application of the Remainder Theorem and the Factor Theorem.
Since f(x) divided by (x + 1) leaves a remainder of 10 the Remainder Theorem tells us that f(1) = 10.
Since f(x) divided by (x  2) leaves a remainder of 4 the Remainder Theorem tells us that f(2) = 4.
Since f(1) = 0 the Factor Theorem tell us that (x  1) is a factor of f(x)
Since f(2) = 0 the Factor Theorem tell us that (x + 2) is a factor of f(x)
Hence f(x) = (x  1)(x + 2) g(x) for some polynomial g(x). But f(x) is a third degree polynomial and hence g(x) is a first degree polynomial, that is g(x) = ax + b for some numbers a and b.
Can you complete the problem now?
Penny
Meadow wrote back,
Hi. I understand that using the remainder and factor theorems you get
f(x) = (x1)(x+2)(g(x)).
But i still don't understand what you do next. Do you find the exact polynomial f(x) and then divide it by x3? if so how would you find the polynomial? Or, do you not find the polynomial at all and just use x3 in some way? I'm sorry, but i still don't understand. can you please help?
Hi Again,
You know that f(1) = 10 so substitute x = 1 into
f(x) = (x1)(x+2)(ax+b)
and you will get a linear equation in a and b. Likewise f(2) = 4 gives you a second linear equation in a and b. Solve these two equation for a and b.
Once you find a and b you have the exact polynomial
f(x) = (x1)(x+2)(ax+b)
You don't need to divide by (x3) to find find the remainder, you can use the remainder theorem again.
Penny
