Question 1082433
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You are given

x + y = a,           (1)
{{{x^2 + y^2}}} = b           (2)

You are asked to find the value of {{{x^3 + y^3}}}.

Square the equality (1) (both sides). You will get

{{{x^2 + 2xy + y^2}}} = {{{a^2}}}.     (3)

Subtract (2) from (3) (both sides). You will get

2xy = {{{b - a^2}}}.         (4)

Now we are ready to evaluate {{{x^3 + y^3}}}.


  {{{x^3 + y^3}}} = {{{(x+y)*(x^2-xy+y^2)}}}       (I suppose you know this classic/basic factoring)

= {{{a*((x^2 + y^2)- xy))}}}                 (I just replaced the factor x+y by its value a, according to (1))

= {{{a*(b - ((b - a^2)/2))}}}                (I just replaced {{{x^2 + y^2}}} by 'b' and replaced  xy  by  {{{(b-a^2)/2}}}, due to (4) )

= {{{a*(a^2+b)/2}}} = {{{(a^3+ab)/2}}}.
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Solved.