To determine how many zeros the polynomial function f(x) = 7x3 + 12x2 + 16x + 23x + 42 has, we begin by simplifying the function:
Combine like terms:
f(x) = 7x3 + (12x2 + 16x + 23x) + 42
= 7x3 + 12x2 + 39x + 42Next, we can analyze the function to determine the number of zeros. A zero of a function is a value of x such that f(x) = 0.
The degree of the polynomial is 3 (the highest power of x), which indicates that there can be up to 3 real roots (or zeros) for this function. However, determining the exact number of zeros requires additional analysis.
To find the zeros, we may apply the Rational Root Theorem, which suggests checking potential rational roots. In this case, the possible rational roots would be the factors of 42 (the constant term) divided by the factors of 7 (the leading coefficient):
- Possible roots: ±1, ±2, ±3, ±6, ±7, ±14, ±21, ±42
Next, we evaluate the function at these potential roots:
- For x = -2:
f(-2) = 7(-2)3 + 12(-2)2 + 39(-2) + 42 = 7(-8) + 12(4) - 78 + 42 = -56 + 48 - 78 + 42 = -44 (not a root) - Tests for other candidates may also yield no rational roots. Therefore, we can also employ numerical methods or graphing to find approximate roots.
By using tools like graphing calculators or numerical software, we can estimate or visualize the function:
- Graphing the function could show us where the curve intersects the x-axis, giving a visual indication of real roots.
- By applying numerical methods such as Newton’s method, we may find the approximate values of the zeros.
In conclusion, while we can assert the maximum potential for three zeros due to the function’s degree, detailed numerical analysis or graphing is recommended to identify the exact count of real zeros. Based on preliminary investigation, it appears that this cubic polynomial may have:
- 0, 1, 2, or 3 real zeros, depending on its behavior as we analyze further.
So, the answer to your question is that the function can have up to 3 zeros, but an exact count would require further analysis.