Escreva sob a forma de polinómio
Binómio de Newton: Infinito 12 A - Parte 1 Pág. 176 Ex. 53
Enunciado
Escreva sob a forma de polinómio as expressões:
- ${{(2+x)}^{4}}$
- ${{(1-2x)}^{5}}$
- ${{(\sqrt{2}+x)}^{6}}$
- ${{(\sqrt{2}x-\sqrt{3})}^{4}}$
Resolução
- Ora,
$$\begin{array}{*{35}{l}}
{{(2+x)}^{4}} & = & \sum\limits_{k=0}^{4}{{}^{4}{{C}_{k}}\times {{2}^{4-k}}\times {{x}^{k}}} \\
{} & = & {}^{4}{{C}_{0}}\times {{2}^{4}}\times {{x}^{0}}+{}^{4}{{C}_{1}}\times {{2}^{3}}\times {{x}^{1}}+{}^{4}{{C}_{2}}\times {{2}^{2}}\times {{x}^{2}}+{}^{4}{{C}_{3}}\times {{2}^{1}}\times {{x}^{3}}+{}^{4}{{C}_{4}}\times {{2}^{0}}\times {{x}^{4}} \\
{} & = & 4\times 16\times 1+4\times 8\times x+6\times 4\times {{x}^{2}}+4\times 2\times {{x}^{3}}+1\times 1\times {{x}^{4}} \\
{} & = & 64+32x+24{{x}^{2}}+8{{x}^{3}}+{{x}^{4}} \\
\end{array}$$ - Ora,
$$\begin{array}{*{35}{l}}
{{(1-2x)}^{5}} & = & \sum\limits_{k=0}^{5}{{}^{5}{{C}_{k}}\times {{1}^{5-k}}\times {{(-2x)}^{k}}} \\
{} & = & {}^{5}{{C}_{0}}\times {{1}^{5}}\times {{(-2x)}^{0}}+{}^{5}{{C}_{1}}\times {{1}^{4}}\times {{(-2x)}^{1}}+{}^{5}{{C}_{2}}\times {{1}^{3}}\times {{(-2x)}^{2}}+{}^{5}{{C}_{3}}\times {{1}^{2}}\times {{(-2x)}^{3}}+{}^{5}{{C}_{4}}\times {{1}^{1}}\times {{(-2x)}^{4}}+{}^{5}{{C}_{5}}\times {{1}^{0}}\times {{(-2x)}^{5}} \\
{} & = & 1\times 1\times 1+5\times 1\times (-2x)+10\times 1\times 4{{x}^{2}}+10\times 1\times (-8{{x}^{3}})+5\times 1\times 16{{x}^{4}}+1\times 1\times (-32{{x}^{5}}) \\
{} & = & 1-10x+40{{x}^{2}}-80{{x}^{3}}+80{{x}^{4}}-32{{x}^{5}} \\
\end{array}$$ - Ora,
$$\begin{array}{*{35}{l}}
{{(\sqrt{2}+x)}^{6}} & = & \sum\limits_{k=0}^{6}{{}^{6}{{C}_{k}}\times {{(\sqrt{2})}^{6-k}}\times {{x}^{k}}} \\
{} & = & {{(\sqrt{2})}^{6}}+6\times {{(\sqrt{2})}^{5}}x+15\times {{(\sqrt{2})}^{4}}{{x}^{2}}+20\times {{(\sqrt{2})}^{3}}{{x}^{3}}+15\times {{(\sqrt{2})}^{2}}{{x}^{4}}+6\times {{(\sqrt{2})}^{1}}{{x}^{5}}+{{x}^{6}} \\
{} & = & 8+24\sqrt{2}x+60{{x}^{2}}+40\sqrt{2}{{x}^{3}}+30{{x}^{4}}+6\sqrt{2}{{x}^{5}}+{{x}^{6}} \\
\end{array}$$ - Ora,
$$\begin{array}{*{35}{l}}
{{(\sqrt{2}x-\sqrt{3})}^{4}} & = & \sum\limits_{k=0}^{4}{{}^{4}{{C}_{k}}\times {{(\sqrt{2}x)}^{4-k}}\times {{(-\sqrt{3})}^{k}}} \\
{} & = & {{(\sqrt{2}x)}^{4}}+4\times {{(\sqrt{2}x)}^{3}}\times (-\sqrt{3})+6\times {{(\sqrt{2}x)}^{2}}\times {{(-\sqrt{3})}^{2}}+4\times {{(\sqrt{2}x)}^{1}}\times {{(-\sqrt{3})}^{3}}+{{(-\sqrt{3})}^{4}} \\
{} & = & 4{{x}^{4}}-8\sqrt{6}{{x}^{3}}+36{{x}^{2}}-12\sqrt{6}x+9 \\
\end{array}$$
