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Write an equation expressing c explicitly in terms of a and b if a = x + y, b = x^2 + y^2, c = x^3 + y^3.

 Dec 19, 2019
 #1
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a  =  x  +  y        (1) 

b  = x^2  + y^2   (2)  

c = x^3 + y^3        =  ( x + y)  (x^2  - xy   + y^2)   =  ( x + y) (x^2 + y^2 - xy )   (3)

 

Square both sides of   (1)   and we get that

a^2  =  x^2  + 2xy  + y^2

-2xy  = x^2 + y^2   - a^2

-2xy  =  b - a^2

-xy  =  [ b - a^2 ] / 2      (4)

 

Sub  (1), (2)  (4)   into  (3)    and we have that

 

c =  ( a)  ( b  +  [ b - a^2] / 2 )

 

c = (a) ( b + b/2  - a^2/2 )

 

c  = (a) ( (3/2)b - (1/2)a^2  )

 

c =  (3/2)ab  - (1/2)a^3

 

 

cool cool cool

 Dec 19, 2019

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