If x3 + px + q and x3 + qx + p have a common factor, then which of the

1.	If x3 + px + q and x3 + qx + p have a common factor, then which of the following is correct ?(a) p + q = 0 (b) p + q – 1 = 0 (c) (p + q + 1) = 0 (d) p – q + 1 = 0
Answer» (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.

If x3 + px + q and x3 + qx + p have a common factor, then which of the following is correct ?(a) p + q = 0 (b) p + q – 1 = 0 (c) (p + q + 1) = 0 (d) p – q + 1 = 0

Answer»

(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.