Give f(x)=x^5 −5x^4 +4x^3 −3x^2 +2x−1,& α= (2)^(1/3) (((5+3(√3)))^(1/3) − ((2)^(1/3) /( ((5+3(√3)))^(1/3) ))).Find f(α)?


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Question Number 144327 by SOMEDAVONG last updated on 24/Jun/21

$Give f (x) = x^{5} - {5 x}^{4} + {4 x}^{3} - {3 x}^{2} + 2 x - 1, &$ $α = \sqrt[3]{2} (\sqrt[3]{5 + 3 \sqrt{3}} - \frac{\sqrt[3]{2}}{\sqrt[3]{5 + 3 \sqrt{3}}}) . Find f (α) ?$
	Answered by Rasheed.Sindhi last updated on 25/Jun/21
	$α= (2)^(1/3) (((5+3(√3)))^(1/3) − ((2)^(1/3) /( ((5+3(√3)))^(1/3) )))(clearly a real number) Simplification of α α^3 =2{(5+3(√3))−(2/(5+3(√3)))−3(2)^(1/3) (((5+3(√3)))^(1/3) − ((2)^(1/3) /( ((5+3(√3)))^(1/3) )))} α^3 =2{(5+3(√3))−(2/(5+3(√3)))−3α} α^3 =2(5+3(√3))−(4/(5+3(√3)))−6α α^3 +6α=2(5+3(√3))−(4/(5+3(√3)))×((5−3(√3))/(5−3(√3))) =10+6(√3)−((20−12(√3))/(25−9(3))) =10+6(√3)+10−6(√3)=20 α^3 +6α−20=0 (α−2)(α^2 +2α+10)=0 α=2 ∣ α=((−2±(√(4−40)))/2)=−1±3i But α is real so −1±3i are discarded. ∴ α=(2)^(1/3) (((5+3(√3)))^(1/3) − ((2)^(1/3) /( ((5+3(√3)))^(1/3) )))=2 f(x)=x^5 −5x^4 +4x^3 −3x^2 +2x−1 f(α)=f(2) =(2)^5 −5(2)^4 +4(2)^3 −3(2)^2 +2(2)−1 =32−80+32−12+4−1 =68−93=−25 f(α)=−25 (please confirm the answer)$
	$α = \sqrt[3]{2} (\sqrt[3]{5 + 3 \sqrt{3}} - \frac{\sqrt[3]{2}}{\sqrt[3]{5 + 3 \sqrt{3}}}) (c l e a r l y a r e a l n u m b e r)$ $S i m p l i f i c a t i o n o f α$ $α^{3} = 2 {(5 + 3 \sqrt{3}) - \frac{2}{5 + 3 \sqrt{3}} - 3 \sqrt[3]{2} (\sqrt[3]{5 + 3 \sqrt{3}} - \frac{\sqrt[3]{2}}{\sqrt[3]{5 + 3 \sqrt{3}}})}$ $α^{3} = 2 {(5 + 3 \sqrt{3}) - \frac{2}{5 + 3 \sqrt{3}} - 3 α}$ $α^{3} = 2 (5 + 3 \sqrt{3}) - \frac{4}{5 + 3 \sqrt{3}} - 6 α$ $α^{3} + 6 α = 2 (5 + 3 \sqrt{3}) - \frac{4}{5 + 3 \sqrt{3}} \times \frac{5 - 3 \sqrt{3}}{5 - 3 \sqrt{3}}$ $= 10 + 6 \sqrt{3} - \frac{20 - 12 \sqrt{3}}{25 - 9 (3)}$ $= 10 + 6 \sqrt{3} + 10 - 6 \sqrt{3} = 20$ $α^{3} + 6 α - 20 = 0$ $(α - 2) (α^{2} + 2 α + 10) = 0$ $α = 2 ∣ α = \frac{- 2 \pm \sqrt{4 - 40}}{2} = - 1 \pm 3 i$ $B u t α i s r e a l s o - 1 \pm 3 i a r e d i s c a r d e d .$ $∴ α = \sqrt[3]{2} (\sqrt[3]{5 + 3 \sqrt{3}} - \frac{\sqrt[3]{2}}{\sqrt[3]{5 + 3 \sqrt{3}}}) = 2$ $f (x) = x^{5} - {5 x}^{4} + {4 x}^{3} - {3 x}^{2} + 2 x - 1$ $f (α) = f (2)$ $= {(2)}^{5} - 5 {(2)}^{4} + 4 {(2)}^{3} - 3 {(2)}^{2} + 2 (2) - 1$ $= 32 - 80 + 32 - 12 + 4 - 1$ $= 68 - 93 = - 25$ $f (α) = - 25$ $(p l e a s e c o n f i r m t h e a n s w e r)$
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