Algebraic expression

Asked by borahbhairabjyoti60 | 25th Apr, 2019, 09:21: PM

Expert Answer:

G i v e n colon space p equals x plus y minus z comma space q equals y plus z minus x space a n d space r equals z plus x minus y
T o space p r o v e colon space p cubed plus q cubed plus r cubed minus 4 left parenthesis x cubed plus y cubed plus z cubed right parenthesis equals 3 left parenthesis p q r minus 4 x y z right parenthesis
W e space k n o w space t h a t colon space
p cubed plus q cubed plus r cubed minus 3 p q r equals left parenthesis p plus q plus r right parenthesis left parenthesis p squared plus q squared plus r squared minus p q minus q r minus p r right parenthesis space......... space left parenthesis i right parenthesis
A f t e r space s u b s t i t u t i n g space t h e space v a l u e s space o f space p comma q space a n d space r space o n space t h e space r i g h t space s i d e space w e space h a v e
p plus q plus r equals x plus y plus z space.......... space left parenthesis i i right parenthesis
p squared plus q squared plus r squared equals 3 x squared plus 3 y squared plus 3 z squared minus 2 x y minus 2 y z minus 2 x z space...... space left parenthesis i i i right parenthesis space space a n d space
p q plus q r plus p r equals negative x squared minus y squared minus z squared plus 2 x y plus 2 y z plus 2 x z space....... space left parenthesis i v right parenthesis
rightwards double arrow p squared plus q squared plus r squared minus p q minus q r minus p r equals 4 x squared plus 4 y squared plus 4 z squared minus 4 x y minus 4 y z minus 4 x z
rightwards double arrow left parenthesis p plus q plus r right parenthesis left parenthesis p squared plus q squared plus r squared minus p q minus q r minus p r right parenthesis equals left parenthesis x plus y plus z right parenthesis left square bracket 4 left parenthesis x squared plus y squared plus z squared minus x y minus y z minus x z right parenthesis right square bracket
space space space equals 4 left parenthesis x plus y plus z right parenthesis left parenthesis x squared plus y squared plus z squared minus x y minus y z minus x z right parenthesis
U sin g space f o r m u l a space left parenthesis i right parenthesis space f o r space x comma space y space a n d space z space w e space h a v e
left parenthesis p plus q plus r right parenthesis left parenthesis p squared plus q squared plus r squared minus p q minus q r minus p r right parenthesis equals 4 left square bracket x cubed plus y cubed plus z cubed minus 3 x y z right square bracket
rightwards double arrow p cubed plus q cubed plus r cubed minus 3 p q r equals 4 left square bracket x cubed plus y cubed plus z cubed minus 3 x y z right square bracket equals 4 left square bracket x cubed plus y cubed plus z cubed right square bracket minus 4 left square bracket 3 x y z right square bracket
rightwards double arrow p cubed plus q cubed plus r cubed minus 4 left parenthesis x cubed plus y cubed plus z cubed right parenthesis equals 3 left parenthesis p q r minus 4 x y z right parenthesis
H e n c e comma space p r o v e d.

Answered by Renu Varma | 26th Apr, 2019, 03:48: PM

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