calculus - The number of distinct real roots of a polynomial ...
Oct 29, 2015 · Incidentally, in a later post on this same polynomial equation [ The number of distinct real roots of a polynomial of degree 4 ], it was remarked by the poster that this problem comes from an …
Counting the Number of Real Roots of A Polynomial
May 4, 2019 · The best way to solve this is to use Sturm's theorem. This gives an algorithm for computing the number of distinct real roots of any polynomial. The Wikipedia page is quite good, but …
How to show that a polynomial does not have real roots?
Jul 26, 2015 · To show that a polynomial has no real roots, we will try to write it as an equation where the sum of some positive numbers equals a strictly negative number. As the sum of positive numbers …
Strange olympiad question, real roots of a polynomial
Oct 22, 2024 · 3 In algebra, the Vieta formulas are a series of results that relate the coefficients of a polynomial to all its roots, so that complex roots are also included.
Are there general methods to compare the largest real roots of two ...
Apr 29, 2025 · Question Is there a general method to compare the largest real roots of two polynomials with integer coefficients, using only their coefficients, when the roots are less than a specific value? …
calculus - Number of real roots of a function, given one of its ...
Mar 1, 2025 · As there is a root of the derivative between any two roots of the function, and given that the function itself has exactly $r$ real roots, then there are at least $r-1$ real roots of the derivative.
Quartic Equation Solution and Conditions for real roots?
There are 9 cases: 4 distinct real roots. 3 distinct real roots with one of them being a double root. 2 distinct double roots both real. Triple root and and a distinct fourth root. Quadruple root. 2 distinct …
Difference between "two equal roots" and "one root"?
Oct 25, 2016 · Why do we emphasise that there are two different but equal roots, as opposed to saying there is only one root? Is there any geometrical intuition for why it would be incorrect to think of these …
The number of distinct real roots of a polynomial of degree 4
May 4, 2016 · If a degree 4 polynomial has 4 real roots, then it must have at least 3 local extremes, so its derivative must have 3 real roots; by repeating this argument, its second derivative must have 2 …
Proving a function has real roots - Mathematics Stack Exchange
I am not interested in finding roots but interested in proving that the function has real roots. Suppose a function $f(x) = x^2 - 1$ This function obviously has real ...