Understanding the Zeros of the Polynomial
A polynomial function is comprised of terms made up of variables raised to whole number powers and constant coefficients. In this case, we are examining the polynomial:
f(x) = 3x^12 + 17x^8 + 11x^4 + 6x + 23
Definition of Zeros
The zeros of a polynomial function are the values of x for which f(x) = 0. These zeros represent the points at which the graph of the polynomial crosses the x-axis.
Analyzing the Polynomial
To determine the number of zeros, we need to consider the general properties of this polynomial function:
- The function is a polynomial of degree 12. This means it can have up to 12 real roots.
- However, to actually find how many zeros exist, we need to explore the behavior of this polynomial.
Finding Real Zeros
To analyze if f(x) has any real zeros, we can apply the following methods:
- Graphing: Plotting the function can often help visualize where it crosses the x-axis.
- Sign Testing: Check the values of f(x) at several points to see if there are sign changes, which indicates the presence of a root between those points.
- Descartes’ Rule of Signs: This rule can help predict the number of positive and negative real roots based on sign changes in the polynomial.
Conclusion
By applying the above methods, specifically through graphing, we can observe that the polynomial f(x) does not cross the x-axis. The function yields only positive values:
For example, at extremes:
- As x approaches negative infinity, f(x) approaches positive infinity.
- As x approaches positive infinity, f(x) also approaches positive infinity.
Additionally, checking values across a reasonable domain suggests that f(x) remains positive. Therefore, we can conclude:
Final Answer
The polynomial function f(x) = 3x^12 + 17x^8 + 11x^4 + 6x + 23 has 0 real zeros.