I need to calculate the number of real solutions in a polynomial equation. From the Fundamental Theorem of Algebra, I know that an equation of degree n has exactly n roots (but, here the complex ones are also included). And I just need to determine how many real ones.
For example, I am asked to the following equation:
From the Fundamental Algebra Theorem, it is known to have 5 roots (real and complex). Next, I have made a sketch of the function and it comes out that it only cuts the X-axis once between zero and one. Then, this equation would have a real solution and 4 complex solutions. But is there any way to solve it analytically without having to graph the function?