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in Polynomials by (24.0k points)
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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

1 Answer

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Best 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.

⇒ α\(\frac{p^2-q^2}{q-p}\) ⇒ 1 = – (p + q) ⇒ p + q + 1 = 0.

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