2026 · Q20
The LCM and HCF of two polynomials p(x) and q(x) are (x+k)²(x²−7x+6) and (x+k) respectively. If [x³+(2k−1)x²+k(k−2)x−k²] is one of the polynomials, then which of the following statements is/are correct?
- IThe other polynomial is (x+k)(x−6) where k ≠ −6.
- II(x+k) cannot be the HCF if k=−6. Select the answer using the code given below:
Elementary MathematicsHCF and LCM
- AI only
- BII only
- CBoth I and II
- DNeither I nor II
Explanation
The given polynomial factors as (x + k)²(x − 1) (x = −k is a root, and dividing out leaves x² + (k − 1)x − k = (x + k)(x − 1)). Then the other polynomial equals LCM × HCF ÷ this polynomial = (x + k)(x − 6), valid when k ≠ −6, so statement I is correct. If k = −6, the stated HCF (x − 6) would not be the true HCF of (x − 6)²(x − 1) and (x − 6)², which is (x − 6)², so statement II is also correct. Hence option C.