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

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To find the remainder when dividing the polynomial f(x) by x² - 4x + 3, which factors into (x-1)(x-3), the remainders from previous divisions by (x-1) and (x-3) are utilized. The remainder can be expressed in the form ax + b. By substituting the known values f(1) = 2 and f(3) = 4 into the equations derived from the polynomial division, two equations can be formed to solve for the coefficients a and b. This method allows for the determination of the remainder when f(x) is divided by x² - 4x + 3. The approach is logical and follows polynomial remainder theorem principles.
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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
 
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What is the degree of the remainder of f(x) divided by x2 - 4x + 3?
 


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

but how do i find a?
 


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)?
 


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
 

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