To determine the number of zeros for the function f(x) = 2x14 + 14x6 + 27x3 + 13x + 12, we can analyze this polynomial by considering its degree and applying the Fundamental Theorem of Algebra.
The degree of the polynomial is determined by the term with the highest exponent, which in this case is 2x14. Therefore, the degree of the polynomial is 14.
According to the Fundamental Theorem of Algebra, a polynomial of degree n can have up to n roots, or zeros. This means that since our function is a 14th-degree polynomial, it can have up to 14 zeros.
Now, to find the actual number of distinct zeros, we can analyze the behavior of the function. We can look for critical points through derivatives or evaluate the function at various points:
- At x = 0: f(0) = 12 (positive)
- As x approaches positive infinity: f(x) also approaches positive infinity
- As x approaches negative infinity: f(x) still approaches positive infinity (since the leading term is positive and has an even degree)
Next, we can check various points:
- f(-1) = 2(-1)14 + 14(-1)6 + 27(-1)3 + 13(-1) + 12 = 2 + 14 – 27 – 13 + 12 = -12 (negative)
- f(-2) = 2(-2)14 + 14(-2)6 + 27(-2)3 + 13(-2) + 12, which is positive (since all terms will yield positive values.)
From this investigation, we can conclude that:
- The function changes from negative to positive between -1 and -2, indicating at least one zero.
- Between any two points where the function changes signs (one positive and one negative), there exists at least one real root due to the Intermediate Value Theorem.
Summarizing:
- The polynomial can have a maximum of 14 zeros.
- Based on initial evaluations and behavior analysis, it’s likely that there are multiple real zeros. However, without a graphical representation or further numerical methods, it’s difficult to pinpoint the exact count.
In conclusion, while the exact number of zeros is not easily determined through simple evaluation, we ascertain that f(x) can have up to 14 zeros, with at least one being present in the examined intervals.