If-x-4-ax-2-bx-c-0-t-4-At-2-B-0-Find-A-and-B-

Question Number 65062 by ajfour last updated on 24/Jul/19

I f x^{4} + {a x}^{2} + b x + c = 0

\Rightarrow t^{4} + {A t}^{2} + B = 0

F i n d A a n d B .

Commented by MJS last updated on 24/Jul/19

I once posted that matrix method by

Tschirnhaus \dots cannot find it right now,

must be somewhere on this forum

it should help in this case

Commented by MJS last updated on 24/Jul/19

found it : question 54775

Commented by MJS last updated on 24/Jul/19

{you}^{'} ll have to adapt it to remove the linear

term but it looks possible

Commented by MJS last updated on 25/Jul/19

\dots you also need to learn how to use the

method with a polynome of 4^{th} degree \dots

maybe you can find something on the web

Commented by ajfour last updated on 25/Jul/19

M j S S i r, t h a n k s f o r y o u r

s i n c e r e s u g g e s t i o n s a n d y e s

m y m e t h o d i s h o p e l e s s l y

c o m p l i c a t e d . .

Answered by ajfour last updated on 26/Jul/19

l e t x = \frac{p t + q}{t + 1}

\Rightarrow p^{4} t^{4} + 4 p^{3} {q t}^{3} + 6 p^{2} q^{2} t^{2} + 4 {p q}^{3} t + q^{4}

+ a (p^{2} t^{2} + 2 p q t + q^{2}) (t^{2} + 2 t + 1)

+ b (p t + q) (t^{3} + 3 t^{2} + 3 t + 1)

+ c (t^{4} + 4 t^{3} + 6 t^{2} + 4 t + 1) = 0

\Rightarrow

(p^{4} + {a p}^{2} + b p + c) t^{4} +

(4 p^{3} q + 2 {a p}^{2} + 2 a p q + 3 b p + b q + 4 c) t^{3}

+ (6 p^{2} q^{2} + {a p}^{2} + 4 a p q + {a q}^{2} +

3 b p + 3 b q + 6 c) t^{2} +

(4 {p q}^{3} + 2 a p q + 2 {a q}^{2} + b p + 3 b q + 4 c) t

+ (q^{4} + {a q}^{2} + b q + c) = 0

s i n c e c o e f f i c i e n t s o f t^{3}, t h a v e

t o b e z e r o, \Rightarrow

4 p^{3} q + 2 {a p}^{2} + 2 a p q + 3 b p + b q + 4 c = 0

\dots . (I)

4 {p q}^{3} + 2 a p q + 2 {a q}^{2} + b p + 3 b q + 4 c = 0

\dots . (I I)

s u b t r a c t i n g \Rightarrow

2 p q (p^{2} - q^{2}) + a (p^{2} - q^{2}) + b (p - q) = 0

A n d a s p \neq q, \Rightarrow

2 p q (p + q) + a (p + q) + b = 0 \dots (i)

A d d i n g (I), (I I)

2 p q (p^{2} + q^{2}) + a (p^{2} + q^{2}) + 2 a p q +

2 b (p + q) + 4 c = 0 \dots . . (i i)

l e t p + q = s, p q = m

t r a n s f o r m i n g (i) & (i i)

________________________

s (2 m + a) + b = 0 \dots . (A)

(2 m + a) (s^{2} - 2 m) +

2 a m + 2 b s + 4 c = 0 \dots . (B)

________________________

(A) \Rightarrow m = - (\frac{a s + b}{2 s})

S u b s t i t u t i n g i n (B)

- \frac{b}{s} (s^{2} + \frac{a s + b}{s}) - \frac{a (a s + b)}{s}

+ 2 b s + 4 c = 0

\Rightarrow \frac{b (a s + b)}{s^{2}} + \frac{a (a s + b)}{s} = b s + 4 c

________________________

\Rightarrow {b s}^{3} + (4 c - a^{2}) s^{2} - 2 a b s - b^{2} = 0

s i s o b t a i n e d f r o m t h i s e q .

m = - (\frac{a s + b}{2 s})

p, q a r e t h e n r o o t s o f q u a d r a t i c

z^{2} - s z + m = 0

________________________

N o w

(p^{4} + {a p}^{2} + b p + c) t^{4} +

+ (6 p^{2} q^{2} + {a p}^{2} + 4 a p q + {a q}^{2} +

3 b p + 3 b q + 6 c) t^{2} +

+ (q^{4} + {a q}^{2} + b q + c) = 0

\Rightarrow

t^{4} + (\frac{6 p^{2} q^{2} + {a p}^{2} + 4 a p q + {a q}^{2} + 3 b p + 3 b q + 6 c}{p^{4} + {a p}^{2} + b p + c}) t^{2}

+ \frac{q^{4} + {a q}^{2} + b q + c}{p^{4} + {a p}^{2} + b p + c} = 0

t^{4} + [\frac{6 p^{2} q^{2} + a {(p + q)}^{2} + 2 a p q + 3 b (p + q)}{p^{4} + {a p}^{2} + b p + c}] t^{2}

+ \frac{q^{4} + {a q}^{2} + b q + c}{p^{4} + {a p}^{2} + b p + + c} = 0

\Rightarrow t^{4} + [\frac{6 m^{2} + {a s}^{2} + 2 a m + 3 b s}{f (p)}] t^{2}

+ \frac{f (q)}{f (p)} = 0

x = \frac{p t + q}{t + 1} .

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