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Suppose $f(x)$ is a polynomial of degree $4$ or greater such that $f(1)=2$, $f(2)=3$, and $f(3)=5$. Find the remainder when $f(x)$ is divided by $(x-1)(x-2)(x-3)$.

michaelcai  Jan 23, 2018
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 #1
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Suppose f(x)  is a polynomial of degree 4 or greater such that f(1)=2,   f(2)=3, and    f(3)=5. Find the remainder when f(x)    is divided by    (x-1)(x-2)(x-3).

 

I don't know the answer but I would like to see someone else discuss it :/

Melody  Jan 24, 2018
edited by Melody  Jan 24, 2018
 #2
avatar+92164 
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I consulted with a higher power and got a great answer!  Thank you  wink

 

Suppose f(x)  is a polynomial of degree 4 or greater such that f(1)=2,   f(2)=3, and    f(3)=5. Find the remainder when f(x)    is divided by    (x-1)(x-2)(x-3).

 

\(\begin{align} \frac{f(x)}{(x-1)(x-2)(x-3)}&=g(x)+\frac{ax^2+bx+c}{(x-1)(x-2)(x-3)}\\ \frac{f(x) \times (x-1)(x-2)(x-3)}{(x-1)(x-2)(x-3)}&=g(x)(x-1)(x-2)(x-3)+\frac{ax^2+bx+c}{(x-1)(x-2)(x-3)}\times (x-1)(x-2)(x-3)\\ f(x)&=g(x)(x-1)(x-2)(x-3)+ax^2+bx+c\\ now\;\;substitute\\ f(1)&=a+b+c=2\\ f(2)&=4a+2b+c=3\\ f(3)&=9a+3b+c=5\\ \end{align} \)

 

So now I have 3 equations and 3 unknowns so I just have to solve them simultaneously.

I solved them using matrices.  These are the unknowns.

 

\(c=2, \;\; b=\frac{-1}{2} \;\;and\;\;a=\frac{1}{2}\\ \text{So when f(x) is divided by (x-1)(x-2)(x-3) the remainder is } \frac{x^2}{2}-\frac{x}{2}+2\)

Melody  Jan 27, 2018
edited by Melody  Jan 27, 2018

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