GATE CS 1987 Q1-xxii — Calculus: Limits, Continuity, Differentiability

We are given the polynomial equation of degree $7$ with real coefficients:

$$f(x) = 7x^{7}+14x^{6}+12x^{5}+3x^{4}+12x^{3}+10x^{2}+5x+7 = 0$$

We can determine the nature of its roots using two fundamental mathematical principles:

1. Complex Conjugate Root Theorem: For any polynomial with real coefficients, non-real complex roots always occur in conjugate pairs. Since the degree of the polynomial is $7$ (an odd number), the maximum number of non-real complex roots it can have is $6$ (i.e., $3$ pairs). Consequently, there must be at least $7 - 6 = 1$ real root.
2. Intermediate Value Theorem (IVT):
   • As $x \to \infty$, the leading term $7x^7 \to \infty$, so $f(x) \to \infty$.
   • As $x \to -\infty$, the leading term $7x^7 \to -\infty$, so $f(x) \to -\infty$.
   Since $f(x)$ is a continuous polynomial function that changes sign from negative to positive over the interval $(-\infty, \infty)$, by the Intermediate Value Theorem, there must exist at least one real value $c$ such that $f(c) = 0$.

Thus, the equation must have at least one real root.