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.

Explanation:

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.

Therefore, we use Descartes' rule of signs to determine the number of positive real roots or negative real roots for the given polynomial function.