a-b-c-d-4-a-2-b-2-c-2-d-2-10-a-3-b-3-c-3-d-3-22-a-4-b-4-c-4-d-4-

Question Number 90793 by kelum last updated on 26/Apr/20

a + b + c + d = 4

a^{2} + b^{2} + c^{2} + d^{2} = 10

a^{3} + b^{3} + c^{3} + d^{3} = 22

a^{4} + b^{4} + c^{4} + d^{4} = ?

Commented by mr W last updated on 26/Apr/20

y o u a r e s u r e t h e q u e s t i o n i s c o r r e c t ?

w e h a v e f o u r u n k n o w n s b u t o n l y

t h r e e e q n ., s o w e c a n n o t g e t a n u n i q u e

s o l u t i o n .

Commented by kelum last updated on 26/Apr/20

Y e s i t s c o r r e c t . I k n o w t h e a n s w e r s

J u s t d o n t k n o w h o w t o g e t t b e

a n s w e r s .

Commented by mr W last updated on 26/Apr/20

w h a t i s t h e a n s w e r ?

i g o t n o u n i q u e s o l u t i o n .

w e c a n o n l y s a y

a^{4} + b^{4} + c^{4} + d^{4} ⩾ 49

Commented by mr W last updated on 29/Apr/20

i f y o u k n o w t h e a n s w e r, p l e a s e s h a r e

i t!

Answered by mr W last updated on 26/Apr/20

p_{1} = a + b + c = 4 - d

p_{2} = a^{2} + b^{2} + c^{2} = 10 - d^{2}

p_{3} = a^{3} + b^{3} + c^{3} = 22 - d^{3}

\Rightarrow e_{1} = 4 - d

p_{2} = e_{1}^{2} - 2 e_{2}

\Rightarrow 10 - d^{2} = {(4 - d)}^{2} - 2 e_{2}

\Rightarrow e_{2} = d^{2} - 4 d + 3

p_{3} = e_{1}^{3} - 3 e_{1} e_{2} + 3 e_{3}

\Rightarrow 22 - d^{3} = {(4 - d)}^{3} - 3 (4 - d) (d^{2} - 4 d + 3) + 3 e_{3}

\Rightarrow 22 - d^{3} = (4 - d) (- 2 d^{2} + 4 d + 7) + 3 e_{3}

\Rightarrow e_{3} = - d^{3} + 4 d^{2} - 3 d - 2

p_{4} = e_{1}^{4} - 4 e_{1}^{2} e_{2} + 4 e_{1} e_{3} + 2 e_{2}^{2} - 4 e_{4}

p_{4} = {(4 - d)}^{4} - 4 {(4 - d)}^{2} (d^{2} - 4 d + 3) + 4 (4 - d) (- d^{3} + 4 d^{2} - 3 d - 2) + 2 {(d^{2} - 4 d + 3)}^{2}

p_{4} = 3 d^{4} - 16 d^{3} + 12 d^{2} + 8 d + 50

a^{4} + b^{4} + c^{4} = 3 d^{4} - 16 d^{3} + 12 d^{2} + 8 d + 50

a^{4} + b^{4} + c^{4} + d^{4} = 4 d (d - 2) (d^{2} - 2 d - 1) + 50 \neq c o n s t a n t

Commented by mr W last updated on 26/Apr/20

o r

p_{1} = a + b + c + d = 4

p_{2} = a^{2} + b^{2} + c^{2} + d^{2} = 10

p_{3} = a^{3} + b^{3} + c^{3} + d^{3} = 22

p_{1} = e_{1} = 4

p_{2} = e_{1}^{2} - 2 e_{2} \Rightarrow 10 = 4^{2} - 2 e_{2} \Rightarrow e_{2} = 3

p_{3} = e_{1}^{3} - 3 e_{1} e_{2} + 3 e_{3} \Rightarrow 22 = 64 - 36 + 3 e_{3} \Rightarrow e_{3} = - 2

p_{4} = e_{1}^{4} - 4 e_{1}^{2} e_{2} + 4 e_{1} e_{3} + 2 e_{2}^{2} - 4 e_{4}

= 4^{4} - 4 \times 4^{2} \times 3 + 4 \times 4 \times (- 2) + 2 \times 3^{2} - 4 e_{4}

= 50 - 4 a b c d

\Rightarrow a^{4} + b^{4} + c^{4} + d^{4} = 50 - 4 a b c d \neq c o n s t a n t

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