While attempting to divide a quartic by a quadratic factor to find the other factors of the given quartic, I can't help feeling I "invented" a way of dividing polynomials.
Suppose you have a quartic polynomial $A$ and a quadratic $B$ where you know that $B$ is a factor of $A$.
You will have:
Now, performing synthetic division is not possible and performing long division is likely to produce mistakes, so I sort of cheated?
I know $B$ goes into $A$ perfectly, which means polynomial $C$ is a quadratic in the form:
Now, I did a sort of cheatery.
I substituted $x=0$ everywhere to obtain:
Then I substituted $x=1$ to get:
And so, you can easily solve for $t,u$ through my method.
My question is whether or not this is a valid way of performing polynomial division, given that the divisor is a factor of the dividend. (or the other way around if I mixed up divisor and dividend.)