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

#2**+2 **

I consulted with a higher power and got a great answer! Thank you

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