use-remainder-theorem-to-factorize-completetly-the-expression-x-3-y-z-y-3-z-x-z-3-x-y-

Question Number 59102 by Tawa1 last updated on 04/May/19

use remainder theorem to factorize completetly the expression

x^{3} (y - z) + y^{3} (z - x) + z^{3} (x - y)

Answered by tanmay last updated on 04/May/19

E = 0 w h e n x = y s o (x - y) i s f a c t o r (E = g i v e n e x p r e s s i o n)

s i m i l a r l y (y - z) a n d (z - x)

x^{3} (y - z) + y^{3} (z - x) + z^{3} (x - y)

= (x - y) (y - z) (z - x) (s y m m e t r i c f u n c t i o n d e g r e e o n e)

= k (x - y) (y - z) (z - x) (x + y + z)

n o w t o f i n d v a l u e o f k

i n e x p r e s s i o n E t h e c o e f f i c i e n t o f x^{3} y = 1

i n d e r i v e d e x p r e s s i o n t h e c o e f f i c u e n t o f

x^{3} y = - k {k x \times y \times - x \times x \to - {k x}^{3} y}

s o a n s w e r

= (- 1) \times (x - y) (y - z) (z - x) (x + y + z)

r e c h e c k

x^{3} (y - z) + y^{3} (z - x) + z^{3} (x - y)

= x^{3} (y - z) + y z (y^{2} - z^{2}) - x (y^{3} - z^{3})

= (y - z) [x^{3} + y^{2} z + {y z}^{2} - {x y}^{2} - x y z - {x z}^{2})

= (y - z) [- x (z^{2} - x^{2}) + y z (z - x) + y^{2} (z - x)]

= (y - z) (z - x) [- x z - x^{2} + y z + y^{2}]

= (y - z) (z - x) [- z (x - y) - (x + y) (x - y)]

= (x - y) (y - z) (z - x) (- x - y - z)

= (- 1) \times (x - y) (y - z) (z + x) (x + y + z)

Commented by Tawa1 last updated on 21/Jul/19

God bless you sir

Facebook Tweet Pin