Such a Fun Math Problem with Descartes' Rule of Signs!

By using Descartes' Rule of Signs in the function f(x) = 7x¹³ - 12x⁹ + 16x⁵ - 23x + 42, we can determine the number of real zeroes in the polynomial equation. The rule helps us count the changes in sign of the coefficients as we move from one term to the next in the polynomial.

Descartes' Rule of Signs states that the number of positive roots (or real zeroes) of a polynomial equation is either equal to the number of variations in sign of the coefficients or less than that by an even number. In the given function f(x) = 7x¹³ - 12x⁹ + 16x⁵ - 23x + 42, we will analyze the changes in sign to determine the number of real zeroes.

Analysis using Descartes' Rule of Signs:

To apply the rule, we observe the signs of the coefficients of the terms of the polynomial.

• From +7x¹³ to -12x⁹: Change from positive to negative.
• From -12x⁹ to +16x⁵: Change from negative to positive.
• From +16x⁵ to -23x: Change from positive to negative.
• From -23x to +42: Change from negative to positive.

Based on the changes in sign, we can see that there are 4 variations (from positive to negative or vice versa) in the polynomial, indicating that there are 4 real zeroes present in the function f(x) = 7x¹³ - 12x⁹ + 16x⁵ - 23x + 42.