1.Nhân đa thức với đa thức: a)(x+1)$^{4}$ .(x+3)$^{4}$ .(x+5)$^{4}$ =? b)(x+1).(x+1)$^{2}$ …..(x+1)$^{99}$ .(x+1)$^{100}$ =? 2.CMR: 50$^{n+1}$ -50

Giải thích các bước giải:

Ta có:

\(\begin{array}{l}
1,\\
a,\\
{\left( {x + 1} \right)^4}.{\left( {x + 3} \right)^4}.{\left( {x + 5} \right)^4}\\
 = {\left[ {\left( {x + 1} \right)\left( {x + 3} \right)\left( {x + 5} \right)} \right]^4}\\
b,\\
\left( {x + 1} \right).{\left( {x + 1} \right)^2}{\left( {x + 1} \right)^3}……..{\left( {x + 1} \right)^{99}}.{\left( {x + 1} \right)^{100}}\\
 = {\left( {x + 1} \right)^{1 + 2 + 3 + …. + 99 + 100}}\\
 = {\left( {x + 1} \right)^{\frac{{\left( {1 + 100} \right).100}}{2}}} = {\left( {x + 1} \right)^{5050}}\\
2,\\
{50^{n + 1}} – {50^n} + {125000^{\frac{1}{3}\left( {n + 1} \right)}}\\
 = {50.50^n} – {50^n} + {\left( {{{50}^3}} \right)^{\frac{1}{3}\left( {n + 1} \right)}}\\
 = {50.50^n} – {50^n} + {50^{n + 1}}\\
 = {50^n}\left( {50 – 1 + 50} \right)\\
 = {50^n}.99 \vdots 11
\end{array}\)