To determine how many zeros the function f(x) = 7x13 + 12x9 + 16x5 + 23x + 42 has, we need to analyze the behavior of the polynomial.
This function is a polynomial of degree 13, which tells us that it can have a maximum of 13 real roots (or zeros) according to the Fundamental Theorem of Algebra. However, not all of these roots need to be real numbers.
Next, we can look at the leading term, which is 7x13. Since the leading coefficient (7) is positive and the degree is odd, we know that the ends of the graph will behave in such a way that as x approaches positive or negative infinity, f(x) will approach positive infinity and negative infinity, respectively. This suggests that there will be at least one real root.
Now, we examine the function at certain key points:
- f(0) = 42: this is positive.
- f(1) = 7(1)13 + 12(1)9 + 16(1)5 + 23(1) + 42 = 7 + 12 + 16 + 23 + 42 = 100: this is also positive.
- f(-1) = 7(-1)13 + 12(-1)9 + 16(-1)5 + 23(-1) + 42 = -7 + 12 – 16 – 23 + 42 = 8: still positive.
- f(-2) = 7(-2)13 + 12(-2)9 + 16(-2)5 + 23(-2) + 42: calculating this shows a negative result.
Based on this analysis, we see that there is a sign change between x = -2 and x = -1, indicating that there is at least one real root in that interval. Since the degree of the polynomial is 13, there can be other real roots, but without further computational methods (like numerical solutions or graphical estimation), we can’t definitively count them just by plugging in integers.
In conclusion, while we can confirm there is at least one real root due to the behavior of the function, determining the exact number of zeros without further analysis or tools becomes complex. Therefore, the polynomial likely has at least one real zero, but could have more, and the exact count would require advanced methods like Descartes’ rule of signs or numerical approximations to accurately find all the roots.