if-a-b-s-and-a-3-b-3-t-find-a-2-b-2-and-a-4-b-4-interms-of-s-and-t-

Question Number 49642 by maxmathsup by imad last updated on 08/Dec/18

i f a + b = s a n d a^{3} + b^{3} = t f i n d a^{2} + b^{2} a n d a^{4} + b^{4} i n t e r m s o f s a n d t .

Answered by mr W last updated on 08/Dec/18

a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2}) = (a + b) [{(a + b)}^{2} - 3 a b]

t = s (s^{2} - 3 a b)

\Rightarrow a b = \frac{1}{3} (s^{2} - \frac{t}{s})

a^{2} + b^{2} = a^{2} + 2 a b + b^{2} - 2 a b = {(a + b)}^{2} - 2 a b

= s^{2} - \frac{2}{3} (s^{2} - \frac{t}{s}) = \frac{1}{3} (s^{2} + \frac{2 t}{s})

a^{4} + b^{4} = {(a^{2} + b^{2})}^{2} - 2 {(a b)}^{2} = \frac{1}{9} {(s^{2} + \frac{2 t}{s})}^{2} - \frac{2}{9} {(s^{2} - \frac{t}{s})}^{2}

= \frac{1}{9} (s^{4} + \frac{4 t^{2}}{s^{2}} + 2 s t) - \frac{2}{9} (s^{4} + \frac{t^{2}}{s^{2}} - 2 s t)

= \frac{1}{9} (\frac{2 t^{2}}{s^{2}} + 6 s t - s^{4})

Commented by maxmathsup by imad last updated on 08/Dec/18

t h a n k y o u s i r .