Resposta :
Quadrado da soma:
[tex](\text{a}+\text{b})^2=(\text{a}+\text{b})\cdot(\text{a}+\text{b})[/tex], com [tex]\text{a}, \text{b}\in\mathbb{R}[/tex].
Donde, segue:
[tex](\text{a}+\text{b})^2=\text{a}^2+\text{ab}+\text{ba}+\text{b}^2[/tex]
[tex](\text{a}+\text{b})^2=\text{a}^2+2\text{ab}+\text{b}^2[/tex]
Quadrado da diferença:
[tex](\text{a}-\text{b})^2=(\text{a}-\text{b})\cdot(\text{a}-\text{b})[/tex], com [tex]\text{a}, \text{b}\in\mathbb{R}[/tex].
Donde, segue:
[tex](\text{a}-\text{b})^2=\text{a}^2-\text{ab}+\text{ba}+\text{b}^2[/tex]
[tex](\text{a}-\text{b})^2=\text{a}^2-2\text{ab}+\text{b}^2[/tex]
Produto de uma soma por uma diferença:
[tex](\text{a}+\text{b})\cdot(\text{a}-\text{b})[/tex], com [tex]\text{a}, \text{b}\in\mathbb{R}[/tex].
Donde, segue:
[tex](\text{a}+\text{b})\cdot(\text{a}-\text{b})=\text{a}^2-\text{ab}+\text{ab}-\text{b}^2[/tex]
[tex](\text{a}+\text{b})\cdot(\text{a}-\text{b})=\text{a}^2-\text{b}^2[/tex]
Cubo de uma soma:
[tex](\text{a}+\text{b})^3=(\text{a}+\text{b})\cdot(\text{a}+\text{b})\cdot(\text{a}+\text{b})[/tex], com [tex]\text{a}, \text{b}\in\mathbb{R}[/tex].
Donde, temos:
[tex](\text{a}+\text{b})^3=\text{a}^3+2\text{a}^2\text{b}+\text{ab}^2+\text{a}^2\text{b}+2\text{ab}^2+\text{b}^3[/tex]
[tex](\text{a}+\text{b})^3=\text{a}^3+3\text{a}^2\text{b}+3\text{ab}^2+\text{b}^3[/tex]
Cubo de uma diferença:
[tex](\text{a}-\text{b})^3=(\text{a}-\text{b})\cdot(\text{a}-\text{b})\cdot(\text{a}-\text{b})[/tex], com [tex]\text{a}, \text{b}\in\mathbb{R}[/tex].
Donde, temos:
[tex](\text{a}-\text{b})^3=\text{a}^3-2\text{a}^2\text{b}+\text{ab}^2-\text{a}^2\text{b}+2\text{ab}^2-\text{b}^3[/tex]
[tex](\text{a}-\text{b})^3=\text{a}^3-3\text{a}^2\text{b}+3\text{ab}^2-\text{b}^3[/tex]
Só faltou essa:
[tex](x+y+z)^{3}= x^{2} + y^{2} + z^{2} + 2xy + 2xz + 2yz =)[/tex]