What does Descartes' rule of signs tell you about the real roots of the polynomial?
Real roots of the polynomial mean zeros of the polynomial in real numbers.
Answer: We use Descartes' rule of signs to determine the number of positive real roots or negative real roots for the given polynomial function.
Let us understand Descartes' rule of signs.
Let f(x) be a single-variable polynomial function arranged by descending variable exponent. Descartes's rule of signs states the following:
- The number of positive real roots is either equal to the number of changes in the sign between two consecutive (non-zero) coefficients or less than it by an even number.
- The number of negative real roots is either equal to the number of changes in the sign of coefficients of f(-x) or less than it by an even number.
Let us understand this by example. Consider f(x) = x5 + 4x4 - x3 + 2x2 - 9 arranged in descending variable exponent order.
Observe that there is a 3 times change in the signs of coefficient in this polynomial function. So, this function f(x) has either 3 or 1 positive real root(s).
Now, f(-x) = - x5 + 4x4 + x3 + 2x2 - 9
Here, there is a 2 times change in the signs of coefficient in f(-x). So, this function f(x) has either 2 or 0 negative real roots.