If $HCF$ & $LCM$ of two quadratic expression are $(x-5)$ and $x^3-19x-30$ then find the two expressions.
- $p(x) = (x-5) (x+2) and\ q(x) = (x-5)(x+3)$
- $p(x) = (x-2) (x-8)\ and\ q(x) = (x-1)(x-3)$
- $p(x) = (x+1) (x-2)\ and\ q(x) = (x-1)(x-3)$
- $p(x) = (x-1) (x-2)\ and\ q(x) = (x-1)(x-8)$
The HCF is (x-5). The LCM is x^3 - 19x - 30. Factoring the LCM: (x-5)(x^2+5x+6) = (x-5)(x+2)(x+3). The two expressions must be formed by distributing these factors such that each contains the HCF (x-5). Option A gives (x-5)(x+2) and (x-5)(x+3), which share (x-5) as HCF and their LCM is (x-5)(x+2)(x+3) = x^3 - 19x - 30.
The product of the two quadratic expressions equals the product of their highest common factor and least common multiple, which is (x - 5)(x^3 - 19x - 30). Factoring the cubic part gives x^3 - 19x - 30 = (x + 2)(x + 3)(x - 5). Since both expressions must share the factor (x - 5) and be quadratic, the remaining factors are distributed to give p(x) = (x - 5)(x + 2) and q(x) = (x - 5)(x + 3).