Problem 78 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Show that a polynomial of an odd degree has at least one real root.

Step-by-Step Solution

Answer

In order to show that a polynomial of an odd degree has at least one real root, we first understand that an odd-degree polynomial takes the form: \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\), where \(n\) is odd and the coefficients are real constants. We then analyze the end behavior of the polynomial as \(x\) approaches positive and negative infinity to see that the polynomial takes very large positive and very large negative values. Since the polynomial is continuous for all real numbers, we can apply the Intermediate Value Theorem (IVT), which states that if a continuous function takes two different values, then it must take any value between those two values. Thus, by IVT, there must be a point \(c\) in the domain of the function where \(P(c) = 0\). This concludes that a polynomial of odd degree has at least one real root.

Step 1: Understanding Odd Degree Polynomials

An odd-degree polynomial takes the form: \[P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\] where \(n\) is odd and \(a_n, a_{n-1},\dots, a_1, a_0\) are real constants. As \(n\) is odd, the highest degree term dominates the behavior of the polynomial for large values of \(x\). Specifically, we're interested in the end behavior as \(x\) approaches positive or negative infinity.

Step 2: Determine the end behavior of the polynomial

We analyze the behavior of the polynomial as \(x\) approaches positive and negative infinity: 1. As \(x\) approaches positive infinity \((x \to \infty)\), if \(a_n > 0\), then \(P(x) \to \infty\). If \(a_n < 0\), then \(P(x) \to -\infty\). 2. As \(x\) approaches negative infinity \((x \to -\infty)\), if \(a_n > 0\), then \(P(x) \to -\infty\). If \(a_n < 0\), then \(P(x) \to \infty\). We know that the polynomial takes very large positive and very large negative values, depending on the sign of the leading coefficient and the direction in which \(x\) approaches infinity.

Step 3: Applying the Intermediate Value Theorem (IVT)

The polynomial \(P(x)\) is continuous for all real numbers, as it is a polynomial function. Since the function takes very large positive and very large negative values, the IVT guarantees that it must take any value between those extremes, specifically including zero. Therefore, there must be a point \(c\) in the domain of the function where \(P(c) = 0\), which means the polynomial of odd degree has at least one real root.

Key Concepts

Odd Degree Polynomial

An odd degree polynomial is a polynomial where the highest power of the variable is an odd number. This means the polynomial has a form such as \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\), where \(n\) is odd and all the coefficients \(a_n, a_{n-1}, \dots, a_1, a_0\) are real numbers. Understanding this structure is key to determining the behavior of the polynomial, especially how it behaves as the variable \(x\) moves towards extreme values.For large values of \(x\), the behavior of an odd degree polynomial is primarily influenced by the term with the highest degree, which is \(a_nx^n\). As \(x\) grows toward positive or negative infinity, this term dominates and will dictate the direction in which \(P(x)\) increases or decreases. This aspect is essential when we consider the polynomial's ability to cross the x-axis and thus to have real roots.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental theorem in calculus applicable to continuous functions, like our polynomial functions. This theorem states that if a continuous function, \(f\), takes on two values \(f(a)\) and \(f(b)\) at points \(a\) and \(b\) within an interval, then it must take any value between \(f(a)\) and \(f(b)\) for some point \(c\) in the interval.Applying the IVT to an odd degree polynomial is straightforward. Since odd degree polynomials stretch towards positive infinity in one direction and negative infinity in the other due to their continuous nature, there must be at least one value of \(x\) (a real number) where the polynomial equals zero. This ensures the existence of a real root, anchoring our understanding of why every odd degree polynomial has at least one real root.

Real Roots

A real root of a polynomial is a value of \(x\) where the polynomial equals zero, meaning \(P(x) = 0\). For odd degree polynomials, the assurance of having real roots stems from how these polynomials behave at infinity and the continuous nature of polynomial functions.To visualize, consider the behavior:

• As \(x\) tends towards positive infinity, the polynomial takes very large positive or negative values depending on the leading term's sign.
• Conversely, as \(x\) goes to negative infinity, \(P(x)\) moves towards the opposite extreme.

Because these values cross over zero, the Intermediate Value Theorem guarantees that the polynomial will be zero at least once. Thus, every odd degree polynomial must indeed have at least one real root. This highlights how the polynomial naturally crosses the x-axis, confirming the presence of one or more real roots.