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If-a-2-a-1-0-then-find-a-5-a-4-1-

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Question Number 191553 by MATHEMATICSAM last updated on 25/Apr/23
If a^2 + a + 1 = 0 then find a^5 + a^4 + 1.
Ifa2+a+1=0thenfinda5+a4+1.
Commented by Tinku Tara last updated on 25/Apr/23
a=ω, ω^2 , where ω is cube root of unity.
a=ω,ω2,whereωiscuberootofunity.
Answered by manxsol last updated on 27/Apr/23
(a−1)(a^2 +a+1)=a^3 −1=0 a^3 =1⇒a^2 (a_(=1) ^3 )+a(a_(=1) ^3 )+1=0
(a−1)(a2+a+1)=a3−1=0a3=1⇒a2(a=13)+a(a=13)+1=0
Answered by mr W last updated on 25/Apr/23
a^2 +a+1=0 a^2 =−(a+1) a^4 =a^2 +2a+1=−a−1+2a+1=a a^5 =a^4 ×a=a^2 =−a−1 a^5 +a^4 +1=−a−1+a+1=0 ✓
a2+a+1=0a2=−(a+1)a4=a2+2a+1=−a−1+2a+1=aa5=a4×a=a2=−a−1a5+a4+1=−a−1+a+1=0✓
Answered by Rasheed.Sindhi last updated on 26/Apr/23
Another way... a^2 + a + 1 = 0⇒1=−a^2 −a ▶a^5 + a^4 + 1=a^5 + a^4 −a^2 −a =a^3 (a^2 +a−(1/a) −(1/a^2 )) =a^3 {(a−(1/a))(a+(1/a))+(a−(1/a))} =a^3 (a−(1/a))(a+(1/a)+1) =a^3 (a−(1/a))(((a^2 +a+1)/a)) =a^2 (a−(1/a))(0)=0
Anotherway…a2+a+1=0⇒1=−a2−a▸a5+a4+1=a5+a4−a2−a=a3(a2+a−1a−1a2)=a3{(a−1a)(a+1a)+(a−1a)}=a3(a−1a)(a+1a+1)=a3(a−1a)(a2+a+1a)=a2(a−1a)(0)=0
Answered by Tinku Tara last updated on 05/Jul/23
a=w or w^2 a^5 +a^4 +1=w^2 +w+1=0
a=worw2a5+a4+1=w2+w+1=0

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