If a is not equal to 0, and if x + y = a and x^3 + y^3 = b, write an equation expressing x^2 + y^2 explicitly in terms of a and b.
x+y = a (1)
\(x^3+y^3=b\) (2)
\(x^3+y^3=(x+y)(x^2-xy+y^2)\) (4)
\((x+y)(x^2-xy+y^2)=b\) (3)
\(\frac{b}{(x+y)}=(x^2-xy+y^2)\)
From (1) we know that x+y=a
so
\(\frac{b}{a}=x^2-xy+y^2\) Expressed in terms of a and b as the question asks. and a can't be 0 as given.
\(x^2+y^2=\frac{b}{a}+xy\) Expressed in terms of \(x^2+y^2\)
Notice that
\((x+y)(x^2-xy+y^2)\)
we know \(x^2+y^2=\frac{b}{a}+xy\)
we can subsituite
we also know (x+y)=a
so
\(a*(\frac{b}{a}+xy-xy)\)
Simplify the brackets
\(a*(\frac{b}{a})\)
which is equal to b ,=x^3+y^3 as given.
So this identity is correct that x^2+y^2=b/a+xy