Math Problem Statement

Let p(x) = polynomial anxn+an-1xn-1+...a1x+a0 if an>0 and x is even then which of the following statements is true? 1) x➡️ infinity then f(x)➡️ negative infinity 2) x➡️ negative infinity then f(x)➡️ negative infinity

Solution

Given the polynomial p(x)=anxn+an1xn1++a1x+a0p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 with an>0a_n > 0 and xx even, let's analyze the behavior of p(x)p(x) as xx \to \infty and xx \to -\infty.

  1. limxp(x)\lim_{x \to \infty} p(x)
  2. limxp(x)\lim_{x \to -\infty} p(x)

Since xx is even, consider x=2kx = 2k, where kZk \in \mathbb{Z}.

Analysis for xx \to \infty:

When xx \to \infty, we focus on the leading term anxna_n x^n because it dominates the behavior of the polynomial. Given an>0a_n > 0:

anxna_n x^n \to \infty

So, p(x)p(x) \to \infty as xx \to \infty.

Analysis for xx \to -\infty:

When xx \to -\infty, and since nn is an integer and xx is even, the term xnx^n behaves as:

xn=(x)n=xnx^n = (-|x|)^n = |x|^n

This means that even if xx \to -\infty, xnx^n remains positive. Thus,

anxna_n x^n \to \infty

Hence, p(x)p(x) \to \infty as xx \to -\infty.

Based on these analyses:

  • Statement 1: xx \to \infty then p(x)p(x) \to -\infty is false.
  • Statement 2: xx \to -\infty then p(x)p(x) \to -\infty is false.

Thus, neither statement is true.

Do you want details or have any questions? Here are some questions you might ask next:

  1. Can you explain the behavior of polynomials with odd degree terms?
  2. What happens if the leading coefficient ana_n is negative?
  3. How does the parity (odd/even) of the highest degree term affect the polynomial's behavior at infinity?
  4. What is the significance of the sign of the leading coefficient in polynomials?
  5. Can we analyze the behavior of polynomial functions with complex coefficients?

Tip: Always consider the leading term's degree and coefficient when analyzing polynomial behavior for large values of xx.

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Math Problem Analysis

Mathematical Concepts

Polynomials
Limit of a Function
Even and Odd Functions

Formulas

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Theorems

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Suitable Grade Level

Advanced High School