Question

a+1/a=2 radical din 3
A=a^4+a^3+a^2+a+1/a+1/a^2+1/a^3+1/a^4

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Answer 1

Răspuns:

Explicație pas cu pas:

a+1/a=2√3        notam a+1/a=t

A=a^4+1/a^4+a³+1/a³+a²+1/a²+a+1/a

(a+1/a)²=a²+2+1/a²  ⇒a²+1/a²= (a+1/a)²-2=t²-2

(a+1/a)³=a³+3a·1/a²+3a²·1/a+1/a³   ⇒ a³+1/a³=( a+1/a)³-3( a+1/a)=t³-3t

(a²+1/a²)²=a^4+2+1/a^4  ⇒a^4+1/a^4=(a²+1/a²)²-2=

=(t²-2)²-2=t^4-4²t+4-2=t^4-4t²+2

A=t^4-4t²+2+t³-3t+t²+t-2=t^4+t³-3t²-2t+2=16×9+24√3-36-4√3=

 =144-36+20√3=108+20√3=4(27+5√3)