What is the remainder when $x^{100} -8x^{99}+12x^{98}-3x^{10}+24x^{9}-36x^{8}+3x^{2}-29x + 41$ is divided by: $x^2-8x+12$.

Question

What is the remainder when: $$x^{100} -8x^{99}+12x^{98}-3x^{10}+24x^{9}-36x^{8}+3x^{2}-29x + 41$$ is divided by: $$x^2-8x+12$$

$x^2-8x+12$ $\leftrightarrow (x-2)(x-6)$

This gives me the polynomial: $(x-2)(x-6)k(x) + r(x)$, where $r(x)$ is the remainder.

The expression gives me the remainder when $x=2$ or $x = 6$

By replacing $x$ with $2$ into the original polynomial you get: $$x^{100} - 4x^{100}+3x^{100}-3x^{10}+12x^{10}-9x^{10}+3x^{2}-29x + 41 = 3x^2-29x + 41$$

By plugging in $2$ I get: $12-58 + 41 = -5$.

Hence $r(x)$ should be: $-5$.

This is apparently the wrong answer. What did I do wrong?

Thank you kindly for your help!

Answer 1

First note that $r(x)=ax+b$ for some $a,b\in\mathbb Z$
For $x=2$ you get $r(2)=-5$ (not $r(x)=-5$) $ \Rightarrow 2a+b=-5$. Do the same for $x=6$ to find r(6).Then find $a,b$.

Answer 2

Hint $\ $ You are very close to the solution. By the polynomial division algorithm

$$\begin{eqnarray} &&\rm f(x) &\: =\: &\rm a + bx + (x-2)(x-6)\, g(x)\\ \Rightarrow\ &&\rm f(2) &\:=\:&\rm a + 2b \\ \Rightarrow\ &&\rm f(6) &\:=\:&\rm a + 6b \end{eqnarray}$$

Solving the above two linear equations for $\rm\,a,b\,$ determines the remainder. Note that the system has a unique solution since the determinant $\ne 0,\,$ being the difference of the distinct evaluation points.

Remark $\ $ The same method extends from two points to $\rm\,n\,$ points, i.e. we can uniquely interpolate the degree $\rm < n\,$ remainder through any $\rm\,n\,$ distinct points because a polynomial of degree $\rm < n\,$ over a field is determined uniquely by $\rm\,n\,$ values. Indeed, the above system of equations has Vandermonde determinant equal to the product of the differences of the evaluation points, so it is be nonzero if the roots are distinct. See here for an example.

Answer 3

It is worth noting that $$ \begin{multline} x^{100} -8x^{99}+12x^{98}-3x^{10}+24x^{9}-36x^{8}+3x^{2}-29x + 41\\ =x^{98}(x^2-8x+12)-3x^8(x^2-8x+12)+3(x^2-8x+12)-5x+5 \end{multline} $$ so that the remainder is $-5x+5$.