Bentuk $(a+b)^n$

Buktikan bahwa \[\large \begin{align*} a^2+b^2=(a+b)^2-2ab \end{align*}\]

Penyelesaian:

\[\large \large \begin{align*} (a+b)^2&=a^2+2ab+b^2\\ &=a^2+b^2+2ab\\ (a+b)^2&-2ab=a^2+b^2 \end{align*}\]

Terbukti bahwa  \[\large \large \begin{align*}\color{Orange}{ a^2+b^2=(a+b)^2-2ab} \end{align*}\]

Buktikan bahwa \[\large \begin{align*} a^3+b^3=(a+b)^3-3ab(a+b) \end{align*}\]

Penyelesaian:

\[\large \large \begin{align*} (a+b)^3&=a^3+3a^2b+3ab^2+b^3\\ &=a^3+b^3+3a^2b+3ab^2\\ &=a^3+b^3+3ab(a+b)\\ (a+b)^3&-3ab(a+b)=a^3+b^3 \end{align*}\]

Terbukti bahwa  \[\large \large \begin{align*}\color{DarkRed}{ a^3+b^3= (a+b)^3-3ab(a+b)}\end{align*}\]

Buktikan bahwa \[\large \begin{align*} a^2+b^2+c^2=(a+b+c)^2-2(ab+ac+bc) \end{align*}\]

Penyelesaian:

\[\large \begin{align*} (a+b+c)^2&=(a+\left (b+c \right )^2\\ &=a^2+2a(b+c)+(b+c)^2\\ &=a^2+2ab+2ac+b^2+2bc+c^2\\ &=a^2+b^2+c^2+2ab+2ac+2bc\\ &=a^2+b^2+c^2+2\left (ab+ac+bc \right )\\ (a+b+c)^2&-2\left (ab+ac+bc \right )=a^2+b^2+c^2\\ \end{align*}\]

Terbukti bahwa \[\large \large \begin{align*} \color{DarkGreen} {a^2+b^2+c^2=(a+b+c)^2-2(ab+ac+bc)}\end{align*}\]

Buktikan bahwa \[\large \begin{align*} a^3+b^3+c^3=(a+b+c)^3-3(a+b)(a+c)(b+c)\\ \end{align*}\]

Penyelesaian:

 \[\large \begin{align*} (a+b+c)^3&=(a+(b+c))^3\\ &=a^3+3a^2(b+c)+3a(b+c)^2+(b+c)^3\\ &=a^3+3a^2(b+c)+3a(b+c)^2+b^3+3b^2c+3bc^2+c^3\\ &=a^3+b^3+c^3+3a^2(b+c)+3a(b+c)^2+3bc(b+c)\\ &=a^3+b^3+c^3+3(b+c)\left [a^2+a(b+c)+bc\right ]\\ &=a^3+b^3+c^3+3(b+c)\left [a^2+ab+ac+bc\right ]\\ &=a^3+b^3+c^3+3(b+c)\left [a(a+b)+c(a+b)\right ]\\ &=a^3+b^3+c^3+3(b+c)\left [(a+b)(a+c)\right ]\\ &=a^3+b^3+c^3+3(a+b)(a+c)(b+c)\\ (a+b+c)^3&-3(a+b)(a+c)(b+c)=a^3+b^3+c^3\\ \end{align*}\]

Terbukti bahwa 

\[\large \large \begin{align*} \color{DarkOrange}{ {a^3+b^3+c^3=(a+b+c)^3-3(a+b)(a+c)(b+c)}}\\ \end{align*}\]