Using-the-remainder-theorem-to-factorize-completely-the-expression-x-3-y-z-y-3-z-x-z-3-x-y-

Question Number 14775 by tawa tawa last updated on 04/Jun/17

Using the remainder theorem to factorize completely the expression

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

Answered by ajfour last updated on 04/Jun/17

w e c a n s e e t h a t, i f y = z

0 + z^{3} (z - x) + z^{3} (x - z) = 0

s o, (y - z) o r (z - y) i s a f a c t o r

s i m i l a r l y f o r (x - y) o r (y - x)

a n d f o r (z - x) o r (x - z);

h e n c e : \pm (x - y) (y - z) (z - x) i s a

f a c t o r o f t h e e x p r e s s i o n;

w h a t c a n c o m e n e x t i s

o f d e g r e e o n e, f i r s t l e t u s s e e i f

(x + y + z) h a p p e n s t o b e a f a c t o r

o r n o t, f o r t h a t l e t z = - x - y

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

{(x + y)}^{2} (y^{2} - x^{2})

= 2 x^{3} y + x^{4} - 2 {x y}^{3} - y^{4} +

+ x^{2} y^{2} - x^{4} + 2 {x y}^{3} - 2 x^{3} y + y^{4} - x^{2} y^{2}

= 0

s o t h e e x p r e s s i o n i s

\pm (x - y) (y - z) (z - x) (x + y + z)

t o c h e c k f o r t h e s i g n l e t u s

c o m p a r e c o e f f i c i e n t o f x^{3} y

w e f i n d t h a t t h e y a g r e e w h e n

w e c h o o s e t h e - v e s i g n, s o

f a c t o r i s e d f o r m o f e x p r e s s i o n i n

q u e s t i o n i s

(y - x) (z - y) (x - z) (x + y + z) .

Commented by tawa tawa last updated on 04/Jun/17

God bless you sir .