If \(x^4+x^2y^2+y^4=319\) and \(x^2+xy+y^2=29\), then find the value of \(x^2-xy+y^2\).
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 11
The expression \(x^4+x^2y^2+y^4\) splits into two factors: \(x^4+x^2y^2+y^4=(x^2+xy+y^2)(x^2-xy+y^2)\). So the required factor is the first expression divided by the given factor.
\(x^4+x^2y^2+y^4=(x^2+y^2)^2-x^2y^2=(x^2+y^2+xy)(x^2+y^2-xy)\)
\(319=29\times(x^2-xy+y^2)\)
\(x^2-xy+y^2\displaystyle=\frac{319}{29}=11\)