We can find the complex roots of the function f(x) = x² + 5x³ − 31x² + 19x + 6 by factoring, using numerical methods to estimate potential roots, applying the rational root theorem to find rational roots, and using synthetic division to confirm the roots.
To find the complex roots of the given function, we can use various methods such as factoring, the quadratic formula, or synthetic division. Let's proceed step-by-step to find all the complex roots of the function f(x) = x² + 5x³ − 31x² + 19x + 6.
1. Start by factoring the expression as much as possible. In this case, the function is not easily factorable, so we need to use alternative methods.
2. Use a numerical method or a graphing calculator to estimate the potential roots. By analyzing the graph or using numerical approximation techniques such as the Newton-Raphson method, we can identify some possible roots.
3. Apply the rational root theorem. The rational root theorem states that if a rational number p/q is a root of the function, then p is a factor of the constant term (6 in this case) and q is a factor of the leading coefficient (1 in this case). By testing the factors of 6 (±1, ±2, ±3, ±6) as potential rational roots, we can find any rational roots that exist.
4. Use synthetic division to divide the function by the potential rational roots. By performing synthetic division with each potential rational root, we can determine if it is a root of the function.
5. After performing these steps, we may find that there are no rational roots for the given function. In this case, we can conclude that the roots are complex.