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

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Homework Help Overview

The problem involves finding the remainder when a polynomial f(x) is divided by the quadratic expression x² - 4x + 3, given specific remainders when f(x) is divided by linear factors of that quadratic.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the relationship between the remainders from dividing by (x-1) and (x-3) and the form of the remainder when divided by the quadratic. Questions arise about the degree and form of the remainder, with some suggesting it may be linear.

Discussion Status

The discussion is active, with participants exploring different forms of the remainder and considering how to derive equations from the known remainders. There is no explicit consensus, but several productive lines of reasoning are being pursued.

Contextual Notes

Participants are working under the constraints of the problem's setup, specifically the known remainders when divided by linear factors and the requirement to find the remainder for a quadratic divisor.

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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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