SHTQ: \(C_{10}^k.2^k.x^{10-k}\) có hệ số: \(a_k=C_{10}^k2^k\)
Hệ số lớn nhất khi thỏa mãn:
\(\left\{{}\begin{matrix}a_k\ge a_{k+1}\\a_k\ge a_{k-1}\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}C_{10}^k2^k\ge C_{10}^{k+1}2^{k+1}\\C_{10}^k2^k\ge C_{10}^{k-1}2^{k-1}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{10!}{k!\left(10-k\right)!}\ge\dfrac{10!}{\left(k+1\right)!\left(10-\left(k+1\right)\right)!}.2\\\dfrac{10!}{k!\left(10-k\right)!}.2\ge\dfrac{10!}{\left(k-1\right)!\left(10-\left(k-1\right)\right)!}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{10-k}\ge\dfrac{2}{k+1}\\\dfrac{2}{k}\ge\dfrac{1}{11-k}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}k\ge\dfrac{19}{3}\\k\le\dfrac{22}{3}\end{matrix}\right.\) \(\Rightarrow k=7\)
Hệ số lớn nhất: \(C_{10}^7.2^7\)
