- The polynomial function \(f(x) = \frac{1}{4}(4x^2 + 9)^2(x^2 + 7)\) has no real zeros because its factors \((4x^2 + 9)\) and \((x^2 + 7)\) produce no real solutions when set to zero.
- As a result, the graph has no x-intercepts and does not cross or touch the x-axis.
- The degree of the polynomial is \(6\), which means the maximum number of turning points is \(6 - 1 = 5\).
- The end behavior is determined by the leading term \(4x^6\), so the graph behaves like the power function \(g(x) = 4x^6\) for large values of \(|x|\).
Explanation
- Finding Real Zeros
To find the real zeros of the polynomial function \(f(x) = \frac{1}{4}(4x^2 + 9)^2(x^2 + 7)\), we set \(f(x) = 0\):
\[\frac{1}{4}(4x^2 + 9)^2(x^2 + 7) = 0\]
This equation is satisfied if either factor is zero:
- \((4x^2 + 9)^2 = 0 \implies 4x^2 + 9 = 0 \implies x^2 = -\frac{9}{4}\). Since the square of a real number cannot be negative, there are no real solutions.
- \((x^2 + 7) = 0 \implies x^2 = -7\). Similarly, there are no real solutions for this factor.
Thus, there are no real zeros for the polynomial function \(f(x)\).
- Analyzing x-intercepts
Because there are no real values of \(x\) for which \(f(x) = 0\), the graph of the function never intersects the x-axis. Consequently, there are no x-intercepts, and the graph neither crosses nor touches the x-axis.
- Determining the Maximum Number of Turning Points
To find the maximum number of turning points, we first determine the degree \(n\) of the polynomial. The polynomial is given by:
\[f(x) = \frac{1}{4}(4x^2 + 9)^2(x^2 + 7)\]
Expanding the leading terms:
\[f(x) = \frac{1}{4}(16x^4 + 72x^2 + 81)(x^2 + 7)\]
The term with the highest power of \(x\) is:
\[\frac{1}{4} \cdot 16x^4 \cdot x^2 = 4x^6\]
Thus, the degree of the polynomial is \(n = 6\). The maximum number of turning points for a polynomial of degree \(n\) is \(n - 1\). Therefore, the maximum number of turning points is \(6 - 1 = 5\).
- Determining End Behavior
The end behavior of a polynomial function is governed by its leading term for large values of \(|x|\). We identified the leading term as \(4x^6\). Therefore, the power function that the graph of \(f(x)\) resembles for large values of \(|x|\) is \(g(x) = 4x^6\). As \(|x| \to \infty\), \(f(x)\) behaves like \(4x^6\), meaning both ends of the graph go upward toward positive infinity.
Examples
Polynomial functions are widely used in engineering and physics to model the path of projectiles or the stress distribution in mechanical components. For instance, if an engineer is designing a bridge, they might use a polynomial function to model the shape of a support beam to ensure it remains within specific load-bearing constraints. Understanding the characteristics of polynomials, such as their intercepts and end behavior, helps engineers predict where structural reinforcements are needed and how the component will react under extreme loads.