Meadow
secondary (10-12))
student

hi. can you guys please help me with this problem?

f(x) is a polynomial of degree 3. It leaves a remainder of 10 and 4 when divided by
x + 1 and x - 2 respectively. Given also that f(1) = f(-2) = 0, find the remainder when
f(x) is divided by x - 3.

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) = (x-1)(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 x-3? if so how would you find the polynomial? Or, do you not find the polynomial at all and just use x-3 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) = (x-1)(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) = (x-1)(x+2)(ax+b)

You don't need to divide by (x-3) to find find the remainder, you can use the remainder theorem again.

Penny