2026 · Q7
The LCM and HCF of two polynomials p(x) and q(x) are (x + a)(x^3 - a^3) and (x^2 - ax + a^2), respectively. If p(x) = x^4 + a^2x^2 + a^4, then what is q(x) equal to?
Elementary MathematicsHCF and LCM
- Ax^4 - 2a^2x^2 + a^4
- Bx^4 + 2a^2x^2 + a^4
- Cx^2 - a^2
- Dx^4 - a^2x^2 + a^4
Explanation
For any two polynomials, p(x)·q(x)=LCM×HCF. Here LCM×HCF=(x+a)(x³−a³)(x²−ax+a²)=(x+a)(x−a)(x²+ax+a²)(x²−ax+a²). Since p(x)=x⁴+a²x²+a⁴=(x²−ax+a²)(x²+ax+a²), dividing the product by p(x) leaves q(x)=(x+a)(x−a)=x²−a², option C. The key step is recognising the factorisation of x⁴+a²x²+a⁴ as a difference of squares.