(c) (p + q + 1) = 0.
Let the common factor be x – α, then x = α will make the given expressions zero, i.e.,
α2 + pα + q = 0
α2 + qα + p = 0
Solving by the rule of cross-multiplication, we have
\(\frac{α^2}{p^2-q^2}\) = \(\frac{α}{q-p}\) = \(\frac{1}{q-p}\)
From, last two relation, α = 1.
⇒ α2 = \(\frac{p^2-q^2}{q-p}\) ⇒ 1 = – (p + q) ⇒ p + q + 1 = 0.