alleryThe polynomial x² + 1 has no first-degree factors over the real numbers because:
AThe leading coefficient is one
BThe polynomial degree is two not three
CThe equation x² + 1 = 0 has no real solutions
DIt has only one term involving x²
Answer & Solution
Correct answer: C. The equation x² + 1 = 0 has no real solutions
1. A first-degree factor of p(x) corresponds to a real number a with p(a) = 0.
2. The equation x² + 1 = 0 requires x² = −1, which has no real solution.
3. So no real number makes p(a) = 0, and the factor theorem gives no first-degree real factor.
4. Hence x² + 1 cannot be split into a product of two first-degree real polynomials.
5. The other options relate to surface features and not the no-real-root reason.
_Source: SCERT Kerala Std X Mathematics Part-2, Chapter 9-11 (pp 227-246, 2019 ed.)._
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