Rumus-Rumus Aljabar
a2 + b2 = (a + b)2 − 2ab
a3 + b3 = (a + b)3 − 3ab(a + b)
3(a2 + b2)(a + b) = 2(a3 + b3) + (a + b)3
\(x^2+\dfrac1{x^2}=\left(x+\dfrac1x\right)^2-2\)
BUKTI
\begin{aligned}
\left(x+\dfrac1x\right)^2&=x^2+2x\left(\dfrac1x\right)+\dfrac1{x^2}\\[3.8pt]
&=x^2+2+\dfrac1{x^2}\\[3.8pt]
x^2+\dfrac1{x^2}&=\left(x+\dfrac1x\right)^2-2
\end{aligned}
\(x^3+\dfrac1{x^3}=\left(x+\dfrac1x\right)^3-3\left(x+\dfrac1x\right)\)
BUKTI
\begin{aligned}
\left(x+\dfrac1x\right)^3&=x^3+3x^2\left(\dfrac1x\right)+3x\left(\dfrac1x\right)^2+\left(\dfrac1x\right)^3\\[3.8pt]
&=x^3+3x+3x\left(\dfrac1{x^2}\right)+\dfrac1{x^3}\\[3.8pt]
&=x^3+3x+3\left(\dfrac1x\right)+\dfrac1{x^3}\\[3.8pt]
&=x^3+\dfrac1{x^3}+3\left(x+\dfrac1x\right)\\[3.8pt]
x^3+\dfrac1{x^3}&=\left(x+\dfrac1x\right)^3-3\left(x+\dfrac1x\right)
\end{aligned}
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