What is the Remainder When Dividing a Polynomial by x2 - 4x + 3?

[Kushal]

Homework Statement

when the polynomial f(x) is divided by (x-1) the remainder is 2, and when f(x) is divided by (x-3) the remainder is 4. Find the remainder when f(x) is divided by x² - 4x + 3.

The Attempt at a Solution

i thought of adding the remainders since x² - 4x + 3 = (x-1)(x-3) but that wouldn't sound very logical...:S

 

[Tedjn]

What is the degree of the remainder of f(x) divided by x² - 4x + 3?

 

[Kushal]

ermm well it will be in the form (x + a)

but how do i find a?

 

[Tedjn]

While it is true in this case that the remainder is of the form x + a, I think it is usually the case that the remainder is in the form ax + b. How would you find this exactly? Well, consider that f(x) = (x-1)p(x) + 2 = (x-3)q(x) + 4 = (x-1)(x-3)s(x) + (ax+b). What do you get if you plug in f(1) and f(3)?

 

[Kushal]

i would get 2 equations for ax+b.

then i'll be able to solve for a and b simultaneously...:)

i hope this is correct.

thnks