Descartes Rule of Signs

Introduction

So this is the formal rule:

Descartes’ Rule of Signs tells you the maximum number of positive or negative real roots of a polynomial by counting sign changes in its coefficients.

And the robots explanation using my broken brain.

Look at the signs. Every time the signs flip, that’s one possible positive root. Fewer by 2, 4, 6… are also possible.

If a polynomial has N sign changes, then the number of positive real roots is:

• N,
• or N − 2,
• or N − 4,
• or N − 6,

Step 1

Count the number of changes from one sign to another in the equation. It does not matter if negative to positive or positive to negative so

f(x) = x³ - 7x² - 10x - 8

In this example the sign changes once from +x³ to -7x²

Step 2

Put -x as an input, and count the changes again

f(-x) =  x³ - 7x² - 10x - 8
      = -x³ - 7x² + 10x - 8

In this example the sign changes twice from -7x² to +10x and +10x to -8

Step 3

Make a table to list the possibilities. Remember: the maximum number of real roots is the highest power of the polynomial. In this example the degree was 3.

We count the number of sign changes for positive roots, and then again for negative roots by evaluating f(-x). These give the *maximum* number of positive and negative real roots.

For example:

 | + | - | i |
 | 1 | 2 | 0 |

This part is often not explained well. It took a second video (and the robot) to make it clear.

Once you have the maximum values, you keep adding rows where the total number of roots adds up to the degree of the polynomial (in this case 3):

 | + | - | i |
 | 1 | 2 | 0 |
 | 1 | 0 | 2 |

To make the pattern clearer, it helps to use a larger degree. The table below is the full Descartes Possibility Table for a degree‑10 polynomial with 6 positive sign changes and 4 negative sign changes.

It demonstrates the rules you need to remember.

Positive Roots	Negative Roots	Complex Roots
6	4	0
6	2	2
6	0	4
4	4	2
4	2	4
4	0	6
2	4	4
2	2	6
2	0	8
0	4	6
0	2	8
0	0	10

Rules to Remember

• Whatever the maximum number of positive or negative roots is, the other possible values are found

 by subtracting 2 until you reach 0.

• Complex roots must increase by 2 because complex roots always come in conjugate pairs.
• Every row must add up to the degree of the polynomial.

The table above shows all valid combinations that follow these rules.

The other tutor suggested a much simpler way to think about it, and this is the one that finally made it click for me:

• Take the maximum number of positive roots and just keep subtracting 2.*
• Do the same for the negative roots.*

So for the example above:

• 6 positives → 6, 4, 2, 0
• 4 negatives → 4, 2, 0

Then make sure your table includes all of these possibilities.

Extra Time

So watching Professor Leonard he went on to give this sum.

f(x) = -4x⁷ + x³ - x² + 2

We did the above and the result was

3 or 1 positive results
2 or 0 negative results

It is obvious I guess but with an expression with a degree of 7 this means there must be 2 imaginary results which he described as irreducible quadratic or a quadratic that refuses to break. It has no real roots, so you can’t factor it using real numbers.