A polynomial $p(x)$ gives a remainder of $1$ when divided by $x^{100}$ and a remainder of $2$ when divided by $(x-2)^3$. Evaluate $p(x)$.

By the Remainder Theorem, $p(x)$ can be written as $$p(x) =x^{100}\times h(x)+1 = (x-2)^3 \times g(x) +2 $$

for some polynomials $f(x)$ and $g(x)$, but how do I solve further?

Any help will be appreciated.

  • 1
    $\begingroup$ The best you can do is to find the remainder when $p(x)$ is divided by $x^{100}(x-2)^3$, and therefore be able to evaluate $p$ at $0$ and $2$. You cannot actually find $p(x)$. $\endgroup$ – Arthur May 18 '16 at 10:01
  • $\begingroup$ @Arthur Why is that so? $\endgroup$ – Henry May 18 '16 at 10:05
  • $\begingroup$ Let's say you've found a polynomial $p(x)$ that satisfies your conditions. Then for any polynomial $q(x)$, the polynomial $$p(x) + (x-2)^3x^{100}q(x)$$ will also fulfill your conditions. So, as you can see, there is a lot of room for non-uniqueness. This room ammounts exactly to knowing what the remainder of $p(x)$ is when divided by $(x-2)^3x^{100}$, but nothing else. $\endgroup$ – Arthur May 18 '16 at 10:10
  • $\begingroup$ @Henry where did you find this problem? This was on a test today :o $\endgroup$ – Nikunj May 18 '16 at 10:40
  • $\begingroup$ @Nikunj My teacher gave it to me in a worksheet. Btw, which test did it come in? How do you solve that? $\endgroup$ – Henry May 18 '16 at 11:01

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