- The function has a root between \(-10\) and \(-6\) due to the sign change in the table (\(p(-10)=81, p(-6)=-14\)).
- The function is below the x-axis between \(x=2\) and \(x=8\) because \(p(4)=-24\) and there are no other roots in that interval.
- The function is positive between \(x=8\) and \(x=12\) as the values are consistently positive.
- Final Answers: Zero between \(\boxed{-10 \text{ and } -6}\); Graph position is \(\boxed{\text{below}}\); Positive between \(\boxed{8 \text{ and } 12}\).
Explanation
- Analyzing Roots using the Intermediate Value Theorem
We are given a fourth-degree polynomial function \(p(x)\) with no repeated factors. The table of values provides us with information to understand its behavior. First, we identify the roots of the function. A root occurs where \(p(x) = 0\). From the table, we observe:
\(p(-4) = 0\)
\(p(2) = 0\)
\(p(8) = 0\)
Since it is a fourth-degree polynomial with no repeated factors, it must have exactly four distinct real roots. The Intermediate Value Theorem states that if a continuous function has values of opposite signs at the endpoints of an interval, there must be at least one root within that interval. We check for sign changes in the provided values:
- Between \(x = -10\) (\(y = 81\)) and \(x = -6\) (\(y = -14\)), there is a sign change. Therefore, there must be a zero between \(x = -10\) and \(x = -6\).
- Determining the Graph's Position between Roots
Next, we examine the behavior of the graph between the roots \(x = 2\) and \(x = 8\). The value at \(x = 4\) is \(-24\). Since the polynomial has no repeated roots, it cannot cross the x-axis at any point other than its roots. Because the function is negative at \(x = 4\), it must remain negative throughout the interval \((2, 8)\). Thus, the graph of the function should be drawn below the x-axis for values between 2 and 8.
- Identifying Positive Intervals
Finally, we identify the intervals where the function is positive. By checking the values in the table:
- For \(x = -12\) and \(x = -10\), the values are \(280\) and \(81\), both positive.
- For \(x = 10\) and \(x = 12\), the values are \(126\) and \(400\), both positive.
Given the behavior and the roots, the function is positive for all x-values between 8 and 12. Therefore, the function is positive for x-values in the interval \((8, 12)\).
- Conclusion and Final Answer
In conclusion:
- The function must have a zero between the x-values of -10 and -6.
- Between the x-values of 2 and 8, the graph of the function should be drawn below the x-axis.
- The function must be positive for all x-values between 8 and 12.
The final answers are:
- Zero between: \(\boxed{-10 \text{ and } -6}\)
- Graph position between 2 and 8: \(\boxed{\text{below}}\)
- Positive for x-values between: \(\boxed{8 \text{ and } 12}\)
Examples
In data analysis, tables of values often represent complex physical or economic systems modeled by polynomials. By observing where the output (y-value) transitions from positive to negative, researchers can determine the threshold or equilibrium point (root) of the system. For instance, in an engineering design, identifying the range where a function is positive might indicate where a structural material is under tension, helping engineers optimize safety and performance. This process of interpreting tables mirrors how we solve for roots and signs in this mathematical analysis.