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Given P(x)→-∞ as → +∞ and P(x) → +∞o as → which of the following could be true? P(x) = -x6 + 2x³ - 2x - 7 OP(x) = x² – 12x6 + x² − 12x³ – 2 - - - OP(x) = 5x4 -8x³ + 3x² - 2x - 7 OP(x) = -2x5 - 3x³ + 16x - 3

Given P(x)→-∞ as → +∞ and P(x) → +∞o as → which of the following could be true? P(x) = -x6 + 2x³ - 2x - 7 OP(x) = x² – 12x6 + x² − 12x³ – 2 - - - OP(x) = 5x4 -8x³ + 3x² - 2x - 7 OP(x) = -2x5 - 3x³ + 16x - 3

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Given \( P(x) \rightarrow -\infty \) as \( x \rightarrow +\infty \) and \( P(x) \rightarrow +\infty \) as \( x \rightarrow -\infty \), which of the following could be true? 1. \(\circledcirc \, P(x) = -x^6 + 2x^3 - 2x - 7\) 2. \(\bigcirc \, P(x) = x^7 - 12x^6 + x^4 - 12x^3 - 2\) 3. \(\bigcirc \, P(x) = 5x^4 - 8x^3 + 3x^2 - 2x - 7\) 4. \(\bigcirc \, P(x) = -2x^5 - 3x^3 + 16x - 3\) ### Explanation The problem is asking which polynomial function corresponds to the behavior of \( P(x) \) approaching negative infinity as \( x \) approaches positive infinity, and \( P(x) \) approaching positive infinity as \( x \) approaches negative infinity. - **Option 1**: The leading term is \(-x^6\), which determines the end behavior of the polynomial. - **Option 2**: The leading term is \(x^7\). - **Option 3**: The leading term is \(5x^4\). - **Option 4**: The leading term is \(-2x^5\). The key to solving this is in identifying the sign and degree of the leading term in the polynomial, as it dominates the behavior of the polynomial for very large or very small \( x \).
Transcribed Image Text:Given \( P(x) \rightarrow -\infty \) as \( x \rightarrow +\infty \) and \( P(x) \rightarrow +\infty \) as \( x \rightarrow -\infty \), which of the following could be true? 1. \(\circledcirc \, P(x) = -x^6 + 2x^3 - 2x - 7\) 2. \(\bigcirc \, P(x) = x^7 - 12x^6 + x^4 - 12x^3 - 2\) 3. \(\bigcirc \, P(x) = 5x^4 - 8x^3 + 3x^2 - 2x - 7\) 4. \(\bigcirc \, P(x) = -2x^5 - 3x^3 + 16x - 3\) ### Explanation The problem is asking which polynomial function corresponds to the behavior of \( P(x) \) approaching negative infinity as \( x \) approaches positive infinity, and \( P(x) \) approaching positive infinity as \( x \) approaches negative infinity. - **Option 1**: The leading term is \(-x^6\), which determines the end behavior of the polynomial. - **Option 2**: The leading term is \(x^7\). - **Option 3**: The leading term is \(5x^4\). - **Option 4**: The leading term is \(-2x^5\). The key to solving this is in identifying the sign and degree of the leading term in the polynomial, as it dominates the behavior of the polynomial for very large or very small \( x \).
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