CDS 2026 (I) Elementary Mathematics, question 6

Topic: Algebra

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Q6

If $(x-5)$ is the HCF of $x^{2}-x-p$ and $x^{2}-qx-10$ , then what is the value of $(p+q)$ ?

(a) $-23$
(b) $-17$
(c) $17$
(d) $23$ Correct answer

Solution.

Answer: (d) $23$

Because $x - 5$ divides each polynomial, substitute $x = 5$ .
This gives $25 - 5 - p = 0$ , so $p = 20$ ; and $25 - 5q - 10 = 0$ , so $q = 3$ .
The polynomials become $(x - 5)(x + 4)$ and $(x - 5)(x + 2)$ , confirming that their HCF is exactly $x - 5$ .