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 with and even, let's analyze the behavior of as and .
Since is even, consider , where .
Analysis for :
When , we focus on the leading term because it dominates the behavior of the polynomial. Given :
So, as .
Analysis for :
When , and since is an integer and is even, the term behaves as:
This means that even if , remains positive. Thus,
Hence, as .
Based on these analyses:
- Statement 1: then is false.
- Statement 2: then is false.
Thus, neither statement is true.
Do you want details or have any questions? Here are some questions you might ask next:
- Can you explain the behavior of polynomials with odd degree terms?
- What happens if the leading coefficient is negative?
- How does the parity (odd/even) of the highest degree term affect the polynomial's behavior at infinity?
- What is the significance of the sign of the leading coefficient in polynomials?
- 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 .
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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