Let p and q be non zero integers Consider the polynomial A x x 2 px q It is given that x minus m and...........

Quantitative Aptitude (CDS) Quadratic Equations

Let p and q be non-zero integers. Consider the polynomial A (x ) = x 2 + px + q. It is given that (x − m)

and (x − km) are simple factors of A (x ), where m is a non-zero integer and k is a positive integer, k $\geq$ 2. Which one of the following is correct?

(k + 1)² p² = kq

(k + 1)² q = kp²

k²q= (k + 1)p²

k²p²=(k+1)²q

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Explanation:

We have, A(x) = x² + px + q
$\because$ (x−m) and (x− km) are factors of A(x), then m and km are roots of A(x).
$\therefore$ m + km = − p => m(k + 1) = − p
$=> m=\frac{-p}{k+1}................(i)$
and m.km = q
=> m²k=q
$=> \frac{p^{2}}{(k+1)^{2}}k = q\: \: from\: equation(i)$
=>(k+1)²q =kp²
p
k
=
−
( + 1)
…(i)
and m × km = q
Þ m k q 2 =
Þ p
k
k q
2
2 ( + 1)
× = [from Eq. (i)]
Þ (k + 1) q = kp 2 2