Descartes Rule Problem

Question

Use Descartes Rule to find the possible roots of the polynomial and determine all the roots of the polynomial.

15) Extra Credit

𝑦 = 6𝑥^3- 17𝑥^2+ 14𝑥 − 3

what is descartes rule and how can I use it to solve? Thank you

Answer 1

The Descartes rule of signs tells us that this polynomial can have up to three positive roots, as there are three switches of signs, and no negative roots, as there are no cases of the sign staying the same. That the polynomial is an odd degree guarantees at least one real root.

We can either plot the polynomial for positive x to see where the roots are or look at the factors of the constant term and divide by factors of the leading terms to identify possible rational roots.

Sometimes, it's easiest to just add the coefficients first to see if 1 is a root.

For y = 6x^3 - 17x^2 + 14x - 3, we find that is indeed the case; 6 - 17 + 14 - 3 = 0

Now, use synthetic division to factor

1 | 6 -17 14 -3

6 -11

6 -11 3

With such a large absolute value for the coefficient of the middle term versus the other two, we know that we will have two roots, which will be positive - we can verify if we like by calculating the discriminant, b^2 - 4ac, and seeing that it is positive, (from before, or by looking at this polynomial alternating signs).

We can factor this, then, however we wish (quadratic formula, trying rational roots, plotting).

Clearly, all roots are less than 11/6 - see first two coefficients and the third coefficient having the same sign as the leading coefficient - so 3 is not a root, and the coefficients don't add to 0, so 1 is not a root.

Let's try 3/2

(2x - 3)(3x - 1) - lo and behold, this works. The polynomial can be factored as (x - 1)(2x - 3)(3x -1), and roots are

1, 3/2, 1/3