Jawaban problem Winda

 

Jawaban problem Winda

 

Soal 1 :

Jika $latex x+y=3$ dan $latex x^2 +y^2 =7$. Nilai dari $latex x^4 +y^4$ adalah ...

 

Jawaban :

$latex (x+y)^2 =x^2 +y^2 +2xy$

$latex 3^2 = 7 + 2xy$

$latex xy=1$

 

$latex (x+y)^4=x^4 +4x^3y + 6x^2y^2 + 4xy^3 + y^4$

$latex 3^4 = x^4 + y^4 + 4x^2(xy) + 4y^2(xy) + 6(xy)^2$

$latex 81 = x^4 + y^4 + 4x^2(1) + 4y^2(1) + 6(1)$

$latex 81 = x^4 + y^4 + 4(x^2 + y^2) + 6$

$latex 81 = x^4 + y^4 + 4(7) + 6$

$latex 81 = x^4 + y^4 + 34$

$latex 47 = x^4 + y^4$

 

Soal 2

Diketahui $latex x+ \frac{1}{x}=1$

Hitung : $latex (x+ \frac{1}{x})^2 + (x^2+ \frac{1}{x^2})^2 + (x^3+ \frac{1}{x^3})^2 + \cdot + (x^{27}+ \frac{1}{x^{27}})^2$

Jawaban :

Perhatikan bahwa

$latex (x+ \frac{1}{x})(x+ \frac{1}{x})=x^2+ \frac{1}{x^2} + 2$

$latex (1)(1)=x^2+ \frac{1}{x^2} + 2$

$latex 1-2=x^2+ \frac{1}{x^2}$

$latex -1=x^2+ \frac{1}{x^2}$

 

$latex (x+ \frac{1}{x})(x^2+ \frac{1}{x^2})=x^3+ \frac{1}{x^3} + x+ \frac{1}{x}$

$latex (1)(-1)=x^3+ \frac{1}{x^3} +1$

$latex -1-(1)=x^3+ \frac{1}{x^3}$

$latex -2=x^3+ \frac{1}{x^3}$

 

$latex (x+ \frac{1}{x})(x^3+ \frac{1}{x^3})=x^4+ \frac{1}{x^4} + x^2+ \frac{1}{x^2}$

$latex (1)(-2)=x^4+ \frac{1}{x^4} -1$

$latex -2-(-1)=x^4+ \frac{1}{x^4}$

$latex -1=x^4+ \frac{1}{x^4}$

 

$latex (x+ \frac{1}{x})(x^4+ \frac{1}{x^4})=x^5+ \frac{1}{x^5} + x^3+ \frac{1}{x^3}$

$latex (1)(-1)=x^5+ \frac{1}{x^5} -2$

$latex -1-(-2)=x^5+ \frac{1}{x^5}$

$latex 1=x^5+ \frac{1}{x^5}$

 

Diperoleh pola

$latex x+ \frac{1}{x}=1$

$latex x^2+ \frac{1}{x^2}=-1$

$latex x^3+ \frac{1}{x^3}=( x^2+ \frac{1}{x^2})-( x+ \frac{1}{x})=-2$

$latex x^4+ \frac{1}{x^4}=( x^3+ \frac{1}{x^3})-( x^2+ \frac{1}{x^2})=-1$

$latex x^5+ \frac{1}{x^5}=( x^4+ \frac{1}{x^4})-( x^3+ \frac{1}{x^3})=1$

$latex x^6+ \frac{1}{x^6}=( x^5+ \frac{1}{x^5})-( x^4+ \frac{1}{x^4})=2$

Dan seterusnya

  

Kalau dikuadratkan tentu saja hasilnya

$latex 1+1+4+1+1+4+1+1+4+1+1+4+...+1+1+4$

(sebanyak 27)

$latex =6 \times 9=54$

  

(tidak perlu ditulis semuanya, hanya untuk memperjelas)

 

Untuk Sobat Asimtot yang punya cara lebih cepat atau tips dan trik, silahkan dishare di sini ya.. Di komentar. ..

  

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