I am trying to solve exercises from Mathematical Analysis I by Zorich and I am stuck on a problem about reduction of quartic polynomials:
A fourth-degree polynomial $ax ^ 4 + b x ^ 3 \ldots + e$ can be rearded as a product $a (x ^ 2 + p_1 x + q_1) (x ^ 2 + p_2 x + q_2)$ and can always be brought into the form $\frac{(M_1 + N_1t) (M_2 + N_2t)}{(\gamma t + 1) ^ 2}$ by a substitution $x = \frac{\alpha t + \beta}{\gamma t + 1}$.
I guess the part I do not understand is if we make the substitution specified in the problem, then the degree of $t$ in the resulting numerator and denominator should be the same right? How come that the result is in the desired form?