$\left(x^3+p x^2+2 x-5\right)^{19}\left(x^2+q x-41\right)^8$ $\left(x^4-x^3+x-7\right)^6=$ $x^{97}+3

Question

$\left(x^3+p x^2+2 x-5\right)^{19}\left(x^2+q x-41\right)^8$ $\left(x^4-x^3+x-7\right)^6=$ $x^{97}+391 x^{96}+a_{95} x^{95}+a_{94} x^{94}+\ldots a_1 x+a_0$ be an identity, where $p, q, a_{95} \ldots a_0$ are integers. Compute the smallest positive value of $p$.

Professor PC Thomas Classes · Answer

$$
\begin{aligned}
& \left({ }^{19} c_0\left(x^3ight)^{19}+{ }^{19} c_1\left(p x^2+2 x-5ight)\left(x^3ight)^{18}+\ldotsight) \
& \left({ }^8 c_0 x^{16}+{ }^8 c_1(q x-41) x^{14}+\ldotsight)\left({ }^6 c_0 x^{24}+{ }^6 c_1\left(-x^3+x-1ight) x^{20}+\ldotsight) \
& =x^{97}+391 x^{96}+a_{95} x^{95}+\ldots
\end{aligned}
$$Comparing the coefficient of $x^{96}$ we get
$$
\begin{aligned}
& 19 p+8 q-6=391 \
& \Rightarrow 19 p+8 q=397 	ext { Let } q=19 \lambda+k, 0 \leq k<19 \
& p=\frac{397-8(19 \lambda+k)}{19}=21-8 \lambda-2 \frac{(4 k+1)}{19} \
& \frac{4 k+1}{19} 	ext { must be integer } \Rightarrow k=14, p=15-8 \lambda
\end{aligned}
$$For minimum + ve value of $p, \lambda=1$ and $p=7$