Which of the following terms, when added to the given polynomial, will change the end behavior?

y = –2x7 + 5x6 – 24

–x8

–3x5

5x7

1,000

–300


To determine which term will change the end behavior of the given polynomial, we need to focus on the **degree** of the polynomial. The **end behavior** of a polynomial is dictated by its highest degree term, i.e., the term with the highest power of \(x\).

The given polynomial is:
\[
y = -2x^7 + 5x^6 - 24
\]

The highest degree term here is \(-2x^7\), so the polynomial has degree 7, and its end behavior is determined by this term.

- **If a term with a higher degree than 7 is added**, the highest degree will change, and the end behavior will also change.
- **If a term with a lower degree or a constant is added**, it will not affect the end behavior because the highest power of \(x\) will still be 7.

Now, let's examine the options:

1. **–x⁸**: This term has a higher degree (8), so adding it will change the end behavior.
2. **–3x⁵**: This term has a lower degree (5), so it will not change the end behavior.
3. **5x⁷**: This term has the same degree (7), so it will not change the end behavior (though it will modify the coefficient of the \(x^7\) term).
4. **1,000**: This is a constant term, so it will not change the end behavior.
5. **–300**: This is also a constant term, so it will not change the end behavior.

### Conclusion:
The term that will change the end behavior of the given polynomial is **–x⁸**.

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Do you want further details or have any other questions?

Here are 5 related questions to explore:
1. How do you determine the degree of a polynomial?
2. What is the significance of the leading term in a polynomial?
3. How do constants affect the shape of a polynomial graph?
4. Can two polynomials with different leading coefficients have the same end behavior?
5. How would adding a higher degree term affect the overall graph?

**Tip**: Always check the degree of the highest power when analyzing the end behavior of a polynomial.

Which of the following terms, when added to the given polynomial, will change the end behavior? y = –2x⁷ + 5x⁶ – 24 –x⁸ –3x⁵ 5x⁷ 1,000 –300

Understand how adding terms like –x⁸ or 5x⁷ affects the end behavior of the polynomial y = –2x⁷ + 5x⁶ – 24. This problem focuses on the degree of the polynomial and how the highest degree term dictates the polynomial's end behavior. Suitable for Grades 10-12.

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