Which of the following terms, when added to the given polynomial, will change the end behavior?
[tex]y=-2 x^7+5 x^6-24[/tex]
A. [tex]-x^8[/tex]
B. [tex]-3 x^5[/tex]
C. [tex]5 x^7[/tex]
D. 1,000
E. -300
Answer (1)
The end behavior of a polynomial is determined by its leading term.
Adding a term with a higher degree will change the end behavior.
Adding a term with the same degree but a different sign (when combined with the original leading coefficient) will also change the end behavior.
Therefore, the terms − x 8 and 5 x 7 will change the end behavior of the given polynomial. − x 8 , 5 x 7
Explanation
Understanding End Behavior We are given the polynomial y = − 2 x 7 + 5 x 6 − 24 and asked to determine which of the following terms, when added to the polynomial, will change its end behavior.
The end behavior of a polynomial is determined by its leading term (the term with the highest degree). The given polynomial has a degree of 7 and a leading coefficient of -2. This means that as x approaches infinity, y approaches negative infinity, and as x approaches negative infinity, y approaches infinity.
Conditions for Changing End Behavior To change the end behavior, we need to add a term that either has a higher degree than 7, or has the same degree (7) but with a coefficient that, when combined with -2, results in a different sign.
Analyzing Each Term Let's examine each term:
− x 8 : This term has a degree of 8, which is greater than 7. Therefore, adding this term will change the end behavior.
− 3 x 5 : This term has a degree of 5, which is less than 7. Therefore, adding this term will not change the end behavior.
5 x 7 : This term has a degree of 7, which is the same as the original polynomial. Adding this term to the original polynomial gives ( − 2 x 7 + 5 x 7 ) + 5 x 6 − 24 = 3 x 7 + 5 x 6 − 24 . The new leading term is 3 x 7 , which has a positive coefficient. This will change the end behavior.
1000: This is a constant term (degree 0), which will not change the end behavior.
-300: This is a constant term (degree 0), which will not change the end behavior.
Conclusion Therefore, the terms that will change the end behavior are − x 8 and 5 x 7 .
Examples
Understanding end behavior is crucial in various fields. For instance, in physics, when modeling the trajectory of a projectile, the end behavior helps predict its long-term path. Similarly, in economics, analyzing the end behavior of a cost function can help businesses understand how costs will scale as production increases. By understanding the dominant terms, we can make informed decisions about system behavior in extreme conditions.
