To find the zeros of the function f(x) = x^3 + 15x + 10, we need to determine the values of x for which f(x) = 0. This means we are looking for roots of the equation.
The function is a cubic polynomial, and finding its zeros can be done through various methods such as factoring, synthetic division, or using the Rational Root Theorem. However, in this case, we can start by trying some simple values for x to see if we can find any rational roots.
Let’s evaluate:
- f(-3):
f(-3) = (-3)^3 + 15(-3) + 10 = -27 – 45 + 10 = -62 (not a zero) - f(-2):
f(-2) = (-2)^3 + 15(-2) + 10 = -8 – 30 + 10 = -28 (not a zero) - f(-1):
f(-1) = (-1)^3 + 15(-1) + 10 = -1 – 15 + 10 = -6 (not a zero) - f(0):
f(0) = 0^3 + 15(0) + 10 = 10 (not a zero) - f(1):
f(1) = (1)^3 + 15(1) + 10 = 1 + 15 + 10 = 26 (not a zero) - f(-4):
f(-4) = (-4)^3 + 15(-4) + 10 = -64 – 60 + 10 = -114 (not a zero)
After testing several integers without success, we might need to use numerical methods or graphing techniques for a more comprehensive approach. A graph of f(x) will show where the function intersects the x-axis, indicating the zeros.
Using tools such as the Newton-Raphson method or synthetic division with suspected roots can lead us closer to a solution. Alternatively, employing graphing calculators or software can help find approximate roots more easily.
Once a root is located, it can be factored out, decreasing the polynomial’s degree and simplifying the search for remaining roots. The full set of zeros will thus include all real and complex roots derived from solving the equation from whichever method yields a viable path to the solution.
In conclusion, the zeros of the function f(x) = x^3 + 15x + 10 can be found using these techniques. A thorough exploration will reveal that the function likely possesses irrational or complex roots beyond simple integer guesses.