Problem 5 - Basic Analysis: Introduction to Real Analysis

Question

Suppose \(g(x)\) is a monic polynomial of odd degree \(d\), that is, $$ g(x)=x^{d}+b_{d-1} x^{d-1}+\cdots+b_{1} x+b_{0}, $$ for some real numbers \(b_{0}, b_{1}, \ldots, b_{d-1} .\) Show that there exists \(a K \in \mathbb{N}\) such that \(g(-K)<0 .\) Hint: Make sure to use the fact that \(d\) is odd. You will have to use that \((-n)^{d}=-\left(n^{d}\right)\).

Step-by-Step Solution

Answer

There exists a positive integer \(K\) such that \(g(-K) < 0\) due to the dominance of the negative leading term \(-K^d\).

Step 1: Write Down the Monic Polynomial

The given polynomial is of the form:\[ g(x) = x^d + b_{d-1}x^{d-1} + \ + b_1x + b_0 \]where \(d\) is an odd number and the leading coefficient is 1, making it a monic polynomial.

Step 2: Consider the Polynomial at Negative Values

Let's evaluate the polynomial at \(-K\), where \(K\) is a positive integer. We want to find \(g(-K)\):\[ g(-K) = (-K)^d + b_{d-1}(-K)^{d-1} + \ + b_1(-K) + b_0 \]

Step 3: Analyze the Leading Term

For \(d\) odd, \((-K)^d = -(K^d)\), which is negative:\[ (-K)^d = -K^d \]This term dominates the behavior of \(g(-K)\) for large \(K\) because it has the highest exponent.

Step 4: Evaluate Asymptotic Behavior

As \(K\) gets larger, the term \(-K^d\) will have a larger absolute value than the sum of the remaining terms, which are of lower powers of \(K\):\[ |b_{d-1}(-K)^{d-1} + \ + b_1(-K) + b_0| < K^d \]Therefore, \(g(-K)\) becomes negative for sufficiently large \(K\).

Step 5: Conclusion

Thus, there exists some \(K \in \mathbb{N}\) for which \(g(-K) < 0\), mainly due to the dominance of the leading term \(-K^d\) when \(d\) is odd and \(K\) is sufficiently large.

Key Concepts

Monic Polynomial

A monic polynomial is a type of polynomial where the leading term—the term with the highest degree—has a coefficient of 1. This is significant because it simplifies many algebraic manipulations and calculations. For example, if we have a polynomial written as:

• \[g(x) = x^d + b_{d-1}x^{d-1} + \cdots + b_1x + b_0\]

Here, the term \(x^d\) is the leading term, and its coefficient is 1. Having a monic polynomial makes it easier to understand and predict the overall behavior of the polynomial since the leading term plays such a crucial role in determining the polynomial's growth and direction. Understanding monic polynomials helps us identify dominant terms quickly, leading us to look for specific behaviors like growth, decay, and sign changes as its variables get larger or smaller.

Odd Degree Polynomial

In the realm of polynomials, an odd degree polynomial means the highest power of \(x\) present in the polynomial expression is an odd number, such as 1, 3, 5, etc. This property of polynomials determines some key behaviors:

• First, odd degree polynomials are guaranteed to have at least one real root, owing to the fact that their graph will inevitably cross the x-axis.
• Second, the end behavior of the polynomial’s graph will differ at both ends. For example, as \(x\) approaches negative infinity, \(g(x)\) approaches negative infinity, and as \(x\) approaches positive infinity, \(g(x)\) approaches positive infinity, when the leading coefficient is positive (monic polynomial ensures this when even terms are absent).

Understanding this concept is essential when evaluating a polynomial’s sign at extreme values, which is particularly useful in many mathematical contexts including roots and inequalities.

Leading Term Dominance

Leading term dominance refers to the phenomena where the leading term of a polynomial determines the polynomial’s behavior for extreme values of the independent variable. For a polynomial of degree \(d\), the leading term is \(x^d\) or \(-K^d\) in the case of \(g(-K)\). With odd degree polynomials, this concept is especially important:

• The leading term, which is the highest degree term, grows faster than any other term as \(|x|\) increases. It essentially sets the trend of the polynomial's increase or decrease at large values.
• In our context, for large \(K\), \(-K^d\) dominates because its growth in magnitude is larger compared to other terms of lower powers and consequently, the sign of the polynomial aligns with the sign of this leading term.

This dominance helps us state that for sufficiently large \(K\), \(g(-K)\) would reliably yield a negative result, given that \(d\) is odd.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as its variable approaches infinity or negative infinity. For polynomials, the asymptotic behavior is often governed by the leading term:

• For the polynomial \(g(x) = x^d + b_{d-1}x^{d-1} + \cdots + b_1x + b_0\), when \(x\) is very large or very small, the value of \(g(x)\) is closely approximated by the leading term \(x^d\), thanks to leading term dominance.
• In practical terms, as \(K\) becomes very large, the terms involving lower powers of \(-K\) become negligible in comparison to the dominating term \(-K^d\).

Thus, the asymptotic linear trajectory of \(g(x)\) relies heavily on the odd nature of \(d\), ensuring that for large values of \(K\), the polynomial trend and eventual sign can be predicted with accuracy. This is why \(g(-K)\) assumes a reliably negative value under such conditions.