Using the remainder theorem to factorize completely the expression x^3 (y − z) + y^3 (z − x) + z^3 (x − y)


Question and Answers Forum

All Questions Topic List
Algebra Questions
Previous in All Question Next in All Question
Previous in Algebra Next in Algebra
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

	$God bless you sir .$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum