The function p(x) is a polynomial. If p(–9)=–10 and p(–2)=–7, can you conclude that p(x) is ever equal to 0?

To determine whether the polynomial function $p(x)$ is ever equal to 0 given the values $p(-9) = -10$ and $p(-2) = -7$, we can analyze the implications of these values.

1. Analyze Provided Values

We know:

• $p(-9) = -10$
• $p(-2) = -7$

Both results are negative.

2. Continuous Nature of Polynomials

Polynomials are continuous functions, meaning that they do not have gaps or jumps in their values. If a polynomial takes a negative value at two different points, it's possible that it could cross the x-axis (where $p(x) = 0$) depending on the nature of the polynomial.

3. Intermediate Value Theorem

According to the Intermediate Value Theorem, if a continuous function takes on two values of opposite sign around an interval, it must cross zero at least once in that interval. Since $p(-9)$ and $p(-2)$ are both negative, we conclude that there is no guarantee for a root $p(x) = 0$ between these points.

Conclusion

Based on the information provided, we cannot conclude that $p(x)$ is equal to 0 at any point; in fact, the values given suggest that $p(x)$ is negative within that interval.

Final Answer

No, we cannot conclude that $p(x)$ is ever equal to 0 given that both $p(-9)$ and $p(-2)$ are negative.