The question is as follows: Divide the following Polynomial and place the result into Division Statement.
$$\frac{m^4+n^4}{m^2+n^2}$$
Recently did this on a test and was stumped. A few calulators and classmates later, I'm still stumped.
I know that the division statement is $P(x) = q(x)*d(x) + R$. And I know that the remainder is going to be zero, or 1 degree less than the divisor. In this case, that means a linear remainder.
Trying to use long division did not work, and synthetic division is not possible. Without dividing, I can write the following, just given the information at hand.
$$m^4+n^4=q(x) *(m^2+n^2)+R$$
Any and all help appreciated.
Best Answer
Do you mean something like $$m^4+n^4=(m^2-n^2)(m^2+n^2)+2n^4?$$
That's what you get if you just do long division of $m^2+n^2$ into $m^4+n^4$: you get a quotient of $m^2-n^2$ and a remainder of $2n^4$
$$\require{enclose} \begin{array}{r}\color{red}{m^2}\color{green}{-n^2}\\[-3pt] m^2+n^2\enclose{longdiv}{m^4+n^4}\\[-3pt] \underline{\color{red}{m^4+m^2n^2}}\\[-3pt] -m^2n^2+n^4 \\[-3pt] \underline{\color{green}{-m^2n^2-n^4}}\\[-3pt] 2n^4\end{array}$$
[someone might know how to edit this so it looks better!]