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If-a-3-a-1-0-then-find-the-valueof-a-4-a-3-a-2-2a-1-




Question Number 49123 by Pk1167156@gmail.com last updated on 03/Dec/18
If  a^3 −a−1=0  then find the valueof  a^4 +a^3 −a^2 −2a+1
$$\mathrm{If} \\ $$$$\mathrm{a}^{\mathrm{3}} −\mathrm{a}−\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{valueof} \\ $$$$\mathrm{a}^{\mathrm{4}} +\mathrm{a}^{\mathrm{3}} −\mathrm{a}^{\mathrm{2}} −\mathrm{2a}+\mathrm{1} \\ $$
Commented by behi83417@gmail.com last updated on 03/Dec/18
a^3 =a+1  =a^3 (a+1)−(a+1)^2 +2=a^3 (a^3 )−(a^3 )^2 +2=2 .
$${a}^{\mathrm{3}} ={a}+\mathrm{1} \\ $$$$={a}^{\mathrm{3}} \left({a}+\mathrm{1}\right)−\left({a}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{2}={a}^{\mathrm{3}} \left({a}^{\mathrm{3}} \right)−\left({a}^{\mathrm{3}} \right)^{\mathrm{2}} +\mathrm{2}=\mathrm{2}\:. \\ $$
Answered by mr W last updated on 03/Dec/18
a^3 =a+1  a^4 +a^3 −a^2 −2a+1  =a(a+1)+a^3 −a^2 −2a+1  =a^3 −a+1  =a^3 −a−1+2  =0+2  =2
$${a}^{\mathrm{3}} ={a}+\mathrm{1} \\ $$$$\mathrm{a}^{\mathrm{4}} +\mathrm{a}^{\mathrm{3}} −\mathrm{a}^{\mathrm{2}} −\mathrm{2a}+\mathrm{1} \\ $$$$={a}\left({a}+\mathrm{1}\right)+\mathrm{a}^{\mathrm{3}} −\mathrm{a}^{\mathrm{2}} −\mathrm{2a}+\mathrm{1} \\ $$$$=\mathrm{a}^{\mathrm{3}} −\mathrm{a}+\mathrm{1} \\ $$$$=\mathrm{a}^{\mathrm{3}} −\mathrm{a}−\mathrm{1}+\mathrm{2} \\ $$$$=\mathrm{0}+\mathrm{2} \\ $$$$=\mathrm{2} \\ $$

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