Limits and Continuity | q006.html

(cod: P-43-17-7) Let \(p(x) = a_n x^n + \cdots + a_1 x + a_0\), where \(n \in \mathbb{N}\), be a polynomial function with real coefficients. Select the correct alternative:

Suppose \(n\) is odd. If \(a_n>0\), we have \(\lim\limits_{x \to - \infty} p(x) = -\infty\) and \(\lim\limits_{x \to + \infty} p(x) = + \infty.\) However, if \(a_n<0\), we have \(\lim\limits_{x \to - \infty} p(x) = + \infty\) and \(\lim\limits_{x \to + \infty} p(x) = +\infty\). Therefore, it is not possible to affirm that every polynomial of odd degree with real coefficients has a real root.

Suppose \(n\) is even. If \(a_n>0\), we have \(\lim\limits_{x \to - \infty} p(x) = -\infty\) and \(\lim\limits_{x \to + \infty} p(x) = +\infty\). However, if \(a_n<0\), we have \(\lim\limits_{x \to - \infty} p(x) = + \infty\) and \(\lim\limits_{x \to + \infty} p(x) = -\infty.\) In any case, there exist \(a < b\) such that \(p(a)\) and \(p(b)\) have opposite signs. Since \(p(x)\) is continuous on \(\mathbb{R}\), and therefore on \([a,b]\), it follows from the Intermediate Value Theorem that there exists at least one point \(x_0 \in (a,b)\) such that \(p(x_0) = 0\). In summary, every polynomial of even degree with real coefficients has at least one real root.

Suppose \(n\) is odd. If \(a_n>0\), we have \(\lim\limits_{x \to - \infty} p(x) = \infty\) and \(\lim\limits_{x \to \infty} p(x) = -\infty\). However, if \(a_n<0\), we have \(\lim\limits_{x \to - \infty} p(x) = -\infty\) and \(\lim\limits_{x \to +\infty} p(x) = +\infty\). In any case, there exist \(a < b\) such that \(p(a)\) and \(p(b)\) have opposite signs. Since \(p(x)\) is continuous on \(\mathbb{R}\), and therefore on \([a,b]\), it follows from the Intermediate Value Theorem that there exists at least one point \(x_0 \in (a,b)\) such that \(p(x_0) = 0\). In summary, every polynomial of odd degree with real coefficients has at least one real root.

Suppose \(n\) is odd. If \(a_n>0\), we have \(\lim\limits_{x \to - \infty} p(x) = -\infty\) and \(\lim\limits_{x \to + \infty} p(x) = + \infty\). However, if \(a_n<0\), we have \(\lim\limits_{x \to - \infty} p(x) = +\infty\) and \(\lim\limits_{x \to \infty} p(x) = -\infty\). In any case, there exist \(a < b\) such that \(p(a)\) and \(p(b)\) have opposite signs. Since \(p(x)\) is continuous on \(\mathbb{R}\), and therefore on \([a,b]\), it follows from the Intermediate Value Theorem that there exists at least one point \(x_0 \in (a,b)\) such that \(p(x_0) = 0\). In summary, every polynomial of odd degree with real coefficients has at least one real root.

Suppose \(n\) is even. If \(a_n>0\), we have \(\lim\limits_{x \to - \infty} p(x) = + \infty\) and \(\lim\limits_{x \to + \infty} p(x) = -\infty.\) However, if \(a_n<0\), we have \(\lim\limits_{x \to - \infty} p(x) = -\infty\) and \(\lim\limits_{x \to + \infty} p(x) = + \infty\). In any case, there exist \(a < b\) such that \(p(a)\) and \(p(b)\) have opposite signs. Since \(p(x)\) is continuous on \(\mathbb{R}\), and therefore on \([a,b]\), it follows from the Intermediate Value Theorem that there exists at least one point \(x_0 \in (a,b)\) such that \(p(x_0) = 0\). In summary, every polynomial of even degree with real coefficients has at least one root.