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Question Number 215469 by Abdullahrussell last updated on 08/Jan/25

	Answered by alephnull last updated on 08/Jan/25

	$(\frac{1}{2^{2}} + \frac{1}{3^{3}}) = (\frac{1}{4} + \frac{1}{27})$ $= (\frac{27}{108} + \frac{4}{108})$ $t h e r o o t i s \frac{31}{108}$ $s u m o f c o e f f e c i e n t s = P (1)$ $l e t Q (n) = 108 \cdot P (n)$ $P (1) = \frac{Q (1)}{108}$ $l e t Q (n) = a_{6} n^{6} + a_{5} n^{5} . . . + a_{1} n + a_{0}$ $P (1) = Q (1) / 108$ $d i d y o u p r o v i d e a s p e c i f i c p o l y n o m i a l ?$ $i f n o t t h e s u m d e p e n d s o n Q (n)$
	Commented by Abdullahrussell last updated on 08/Jan/25

	$2^{\frac{1}{2}} + 3^{\frac{1}{3}} = \sqrt{2} + \sqrt[3]{3}$
	Answered by mr W last updated on 08/Jan/25
	$say P(x)=a_0 +a_1 x+a_2 x^2 +...+a_6 x^6 with a_6 =1 P(2^(1/2) +3^(1/3) )=Σ_(n=0) ^6 a_n (2^(1/2) +3^(1/3) )^n =0 (2^(1/2) +3^(1/3) )^1 =2^(1/2) +3^(1/3) (2^(1/2) +3^(1/3) )^2 =2+2×2^(1/2) ×3^(1/3) +3^(2/3) (2^(1/2) +3^(1/3) )^3 =3+2×2^(1/2) +6×3^(1/3) +3×2^(1/2) ×3^(2/3) (2^(1/2) +3^(1/3) )^4 =4+8×2^(1/2) ×3^(1/3) +12×3^(2/3) +12×2^(1/2) +3×3^(1/3) (2^(1/2) +3^(1/3) )^5 =60+4×2^(1/2) +20×3^(1/3) +20×2^(1/2) ×3^(2/3) +15×2^(1/2) ×3^(1/3) +3×3^(2/3) (2^(1/2) +3^(1/3) )^6 =17+24×2^(1/2) ×3^(1/3) +60×3^(2/3) +120×2^(1/2) +90×3^(1/3) +18×2^(1/2) ×3^(2/3) determinant (((n \),1,2^(1/2) ,3^(1/3) ,3^(2/3) ,(2^(1/2) ×3^(1/3) ),(2^(1/2) ×3^(2/3) ),),(((...)^0 ),1,,,,,,a_0 ),(((...)^1 ),,1,1,,,,a_1 ),(((...)^2 ),2,,,1,2,,a_2 ),(((...)^3 ),3,2,6,,,3,a_3 ),(((...)^4 ),4,(12),3,(12),8,,a_4 ),(((...)^5 ),(60),4,(20),3,(15),(20),a_5 ),(((...)^6 ),(17),(120),(90),(60),(24),(18),)) a_0 +2a_2 +3a_3 +4a_4 +60a_5 +17=0 a_1 +2a_3 +12a_4 +4a_5 +120=0 a_1 +6a_3 +3a_4 +20a_5 +90=0 a_2 +12a_4 +3a_5 +60=0 2a_2 +8a_4 +15a_5 +24=0 3a_3 +20a_5 +18=0 solving this equation system we get a_0 =1, a_1 =−36, a_2 =12, a_3 =−6, a_4 =−6, a_5 =0, a_6 =1 (given) sum of all coefficients Σ_(n=0) ^6 a_n =−34 P(x)=x^6 −6x^4 −6x^3 +12x^2 −36x+1$
	$s a y P (x) = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{6} x^{6}$ $w i t h a_{6} = 1$ $P (2^{\frac{1}{2}} + 3^{\frac{1}{3}}) = \underset{n = 0}{\sum^{6}} a_{n} {(2^{\frac{1}{2}} + 3^{\frac{1}{3}})}^{n} = 0$ ${(2^{\frac{1}{2}} + 3^{\frac{1}{3}})}^{1} = 2^{\frac{1}{2}} + 3^{\frac{1}{3}}$ ${(2^{\frac{1}{2}} + 3^{\frac{1}{3}})}^{2} = 2 + 2 \times 2^{\frac{1}{2}} \times 3^{\frac{1}{3}} + 3^{\frac{2}{3}}$ ${(2^{\frac{1}{2}} + 3^{\frac{1}{3}})}^{3} = 3 + 2 \times 2^{\frac{1}{2}} + 6 \times 3^{\frac{1}{3}} + 3 \times 2^{\frac{1}{2}} \times 3^{\frac{2}{3}}$ ${(2^{\frac{1}{2}} + 3^{\frac{1}{3}})}^{4} = 4 + 8 \times 2^{\frac{1}{2}} \times 3^{\frac{1}{3}} + 12 \times 3^{\frac{2}{3}} + 12 \times 2^{\frac{1}{2}} + 3 \times 3^{\frac{1}{3}}$ ${(2^{\frac{1}{2}} + 3^{\frac{1}{3}})}^{5} = 60 + 4 \times 2^{\frac{1}{2}} + 20 \times 3^{\frac{1}{3}} + 20 \times 2^{\frac{1}{2}} \times 3^{\frac{2}{3}} + 15 \times 2^{\frac{1}{2}} \times 3^{\frac{1}{3}} + 3 \times 3^{\frac{2}{3}}$ ${(2^{\frac{1}{2}} + 3^{\frac{1}{3}})}^{6} = 17 + 24 \times 2^{\frac{1}{2}} \times 3^{\frac{1}{3}} + 60 \times 3^{\frac{2}{3}} + 120 \times 2^{\frac{1}{2}} + 90 \times 3^{\frac{1}{3}} + 18 \times 2^{\frac{1}{2}} \times 3^{\frac{2}{3}}$ $\begin{array}{cccccccc} n ∖ & 1 & 2^{\frac{1}{2}} & 3^{\frac{1}{3}} & 3^{\frac{2}{3}} & 2^{\frac{1}{2}} \times 3^{\frac{1}{3}} & 2^{\frac{1}{2}} \times 3^{\frac{2}{3}} \\ {(. . .)}^{0} & 1 & a_{0} \\ {(. . .)}^{1} & 1 & 1 & a_{1} \\ {(. . .)}^{2} & 2 & 1 & 2 & a_{2} \\ {(. . .)}^{3} & 3 & 2 & 6 & 3 & a_{3} \\ {(. . .)}^{4} & 4 & 12 & 3 & 12 & 8 & a_{4} \\ {(. . .)}^{5} & 60 & 4 & 20 & 3 & 15 & 20 & a_{5} \\ {(. . .)}^{6} & 17 & 120 & 90 & 60 & 24 & 18 \end{array}$ $a_{0} + 2 a_{2} + 3 a_{3} + 4 a_{4} + 60 a_{5} + 17 = 0$ $a_{1} + 2 a_{3} + 12 a_{4} + 4 a_{5} + 120 = 0$ $a_{1} + 6 a_{3} + 3 a_{4} + {20 a}_{5} + 90 = 0$ $a_{2} + 12 a_{4} + 3 a_{5} + 60 = 0$ $2 a_{2} + 8 a_{4} + 15 a_{5} + 24 = 0$ $3 a_{3} + 20 a_{5} + 18 = 0$ $s o l v i n g t h i s e q u a t i o n s y s t e m w e g e t$ $a_{0} = 1,$ $a_{1} = - 36,$ $a_{2} = 12,$ $a_{3} = - 6,$ $a_{4} = - 6,$ $a_{5} = 0,$ $a_{6} = 1 (g i v e n)$ $s u m o f a l l c o e f f i c i e n t s \underset{n = 0}{\sum^{6}} a_{n} = - 34$ $P (x) = x^{6} - 6 x^{4} - 6 x^{3} + 12 x^{2} - 36 x + 1$
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