Descartes Rule of Signs
Descartes' rule of signs is a technique/rule that is used to find the maximum number of positive real zeros of a polynomial function. It is further extended to find the maximum number of negative real zeros as well. Thus, Descartes' rule of signs can be used to find the maximum number of imaginary roots (complex roots) as well.
Let us learn Descartes' rule of signs by looking at many examples and also let us prove this rule by finding the actual number of zeros.
What is Descartes' Rule of Signs?
Descartes' rule of signs determines the relationship between the number of positive (or negative) real roots and the number of sign changes of a polynomial function. Here, the sign change refers to either "positive to negative" or "negative to positive". This rule is not helpful in determining the exact number of positive (or negative) real roots though.
Descartes' Rule of Signs Definition
By Descartes' rule of signs, if a polynomial in one variable, f(x) = an xn + an-1xn-1 + an-2xn-2 + ...+ a1x + a0 is arranged in the descending order of the exponents of the variable, then:
- The number of positive real zeros of f(x) is either equal to the number of sign changes in f(x) or less than the number of sign changes by an even number.
- The same rule applies to find the number of negative real zeros as well, but then we count the sign changes of f(-x).
Note that while counting the sign changes, we have to avoid the terms with coefficients to be 0.
Here, "by even number" means, if the number of sign changes of f(x) is 6, then its number of positive real zeros can be 6, 4, 2, or 0. Alternatively, Descartes' rule of signs can be defined as follows:
"A polynomial function f(x) (whose terms are arranged in the descending order of the powers of variable) cannot have more positive real roots than the number of sign changes in it."
Corollary of Descartes' Rule of Signs
As the main Descartes' rule of signs talks about the maximum number of positive real roots, its corollary talks about the maximum number of negative real roots. It says:
"A polynomial function f(x) in standard form cannot have more negative real roots than the number of sign changes in f(-x)."
How to Apply Descartes' Rule of Signs?
Descartes' rule of signs is used to find the maximum number of positive and negative real roots. Before the application of this rule of signs, always write the polynomial in the standard form, i.e., descending order of the exponents of the variable, and avoid the terms with coefficient 0. To find the maximum number of positive real roots, count the number of sign changes in f(x), and to find the maximum number of negative real roots, count the number of sign changes in f(-x). Here is an example.
Example: Find the maximum number of positive and negative real zeros of f(x) = 2x2 + x3 - x - 1.
Solution:
Finding Maximum Number of Positive Real Roots:
Writing the given polynomial in the descending order of exponents, f(x) = x3 + 2x2 - x + 1.
Here, the signs from left to right are:
+ + - +
Here there are two sign changes (one from + to -, and the other from - to +). So the maximum number of positive real roots is 2.
Finding Maximum Number of Positive Real Roots:
For finding the maximum number of negative real roots, find f(-x).
f(-x) = (-x)3 + 2(-x)2 - (-x) + 1 = -x3 + 2x2 + x + 1.
Here, the signs from left to right are:
- + + +
There is only one sign change (from - to +) and hence the maximum number of negative real roots is 1. Since 1 cannot be decreased by an even number (because 1-2 = -1, which is negative), the actual number of negative real roots of f(x) is 1.
Descartes' Rule of Signs Chart
Descartes' rule of signs gives only the possible number of positive and negative real roots but it doesn't give the exact number of roots. So we can construct a chart with the possibilities of the number of positive, real, and imaginary roots. While constructing this chart, we have to keep the following things in mind.
- In the last example, we have seen how to find the maximum number of positive real roots and the maximum number of negative real roots of a polynomial function. The roots which are not real are imaginary (complex roots) and we know that the imaginary roots always occur in pairs (for example if 1 + i is a root then 1 - i is also a root). So the number of positive (or negative) real roots is either equal to the number of sign changes of f(x) (or f(-x)) or less than the number of sign changes by an even number.
- Note that if the number of positive (or negative) real roots is 0 or 1, then that number itself is the actual number of positive (or negative) real roots as it cannot be reduced by an even number anymore further. Hence, if we get 0 or 1 in any column (of positive or negative) of the chart, then do not alter it throughout the chart.
- Also, note that the number of roots of a polynomial function (considering the roots with multiplicities to be independent roots) is equal to the degree of the polynomial. Thus, the number of complex roots can be obtained by subtracting the sum of positive and real roots from the degree of the polynomial.
Based on all these facts, we can construct Descartes' rule of signs chart with all possibilities of positive, negative, and imaginary zeros for the same example (mentioned in the previous section) f(x) = x3 + 2x2 - x + 1. Its degree is 3.
| Number of Positive Real roots | Number of Negative Real roots | Number of Imaginary roots |
|---|---|---|
| 2 | 1 | 0 (= 3 - (2 + 1)) |
| 0 | 1 | 2 (3 - (0 + 1)) |
Descartes' Rule of Signs Proof
Descartes' rule of signs has been given without any proof by a mathematician from French whose name is René Descartes in 1637. Isaac Newton later restated this rule in 1707 but has not given the proof though. But we can verify this rule with an example below by finding the number of zeros by Descartes' rule, then by manual computation of the roots, and comparing both the results.
Example: f(x) = x3 - x2 + x - 1.
Computing Number of Zeros By Descartes' Rule of Signs
Signs of f(x) from left to right are:
+ - + -
There are 3 sign changes in f(x) and hence the maximum number of positive real zeros is 3.
Now, f(-x) = (-x)3 - (-x)2 + (-x) - 1 = -x3 - x2 - x - 1.
The signs from left to right are:
- - - -
The number of sign changes is 0. So the number of negative real roots is 0.
Now, we will construct a table with all possibilities using the steps that are explained in the previous section. Note that the degree of the polynomial function is 3.
| Number of Positive Real roots | Number of Negative Real roots | Number of Imaginary roots |
|---|---|---|
| 3 | 0 | 0 (= 3 - (3 + 0)) |
| 1 | 0 | 2 (= 3 - (1 + 0)) |
Computing Zeros Manually
To find the zeros, set f(x) = 0.
x3 - x2 + x - 1 = 0
x2 (x - 1) + 1 (x - 1) = 0
(x - 1) (x2 + 1) = 0
x - 1 = 0; x2 + 1 = 0
x = 1; x2 = -1
x = 1; x = ± i
Thus, the number of positive real roots = 1; number of negative real roots = 0; and the number of imaginary roots = 2 (i and -i).
This is matching with the last row of the table.
Hence, Descartes' rule of signs is verified.
Important Notes on Descartes' Rule of Signs:
- While Applying Descartes' Rule of Signs, before counting the sign changes (of either f(x) or f(-x)), first arrange the polynomial in the descending order of exponents of the variable. For example, it should be in the order ...., x4, x3, x2, x, and constant.
- Avoid writing terms that have a coefficient to be 0. For example do NOT write x2 - 1 as x2 + 0x - 1 while counting the sign changes.
- Descartes' rule of signs cannot give the actual number of positive or real roots (except in the case of 0 or 1).
- If the number of sign changes of (f(x) or f(-x)) is 0 or 1 then that number itself is the actual number of real roots as it cannot be decreased by an even number anymore.
- While finding f(-x), only the signs of terms of f(x) with odd exponents will change.
☛Related Topics:
Descartes' Rule of Signs Examples
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Example 1: Find the maximum number of positive and negative real zeros of the polynomial function f(x) = 2x4 + x3 - 6x2 - 7x + 1.
Solution:
The polynomial is already in the required standard form and hence we don't need to rearrange the terms.
Finding Maximum Number of Positive Real Zeros:
Count the sign changes in f(x).
The number of sign changes = 2.
So the maximum number of positive real zeros = 2.
Finding Maximum Number of Negative Real Zeros:
f(x) = 2(-x)4 + (-x)3 - 6(-x)2 - 7(-x) + 1 = 2x4 - x3 - 6x2 + 7x + 1.
Count the sign changes in f(x).
The number of sign changes = 2.
So the maximum number of negative real zeros = 2.
Answer: The maximum number of positive real zeros = 2 and the maximum number of negative real zeros = 2.
FAQs on Descartes' Rule of Signs
Which Rule of Signs Can be Used to Find the Number of Positive Real Zeros?
Descartes' rule of signs is used to find the maximum number of positive real zeros of a polynomial function. The maximum number of positive real zeros of f(x) is equal to the number of sign changes when f(x) is written from highest to lowest exponents of variable x. For example, the maximum number of positive real zeros of f(x) = x3 + 3x - 2 is 1 as there is only one sign change here.
State Descartes' Rule of Signs.
For any polynomial function f(x) (whose terms are arranged in the descending order of powers of variable):
- Maximum number of positive real roots = number of sign changes of f(x)
- Maximum number of negative real roots = number of sign changes of f(-x)
The actual number of positive/negative real roots may vary from its maximum number by an even number.
What are the Advantages and Disadvantages of Descartes Rule of Signs?
The Descartes rule of signs is used to find the maximum number of positive/negative real zeros of a polynomial function but it cannot help to find the actual zeros.
What Does Descartes' Rule Say About the Number of Positive Real Roots?
According to Descartes' rule of signs, the number of positive real roots of a polynomial function f(x) is either equal to the number of sign changes in f(x) or less than the number of sign changes by an even number. Note that before applying this rule, arrange the terms of f(x) such that they are in decreasing order of exponents of variable x.
What Does Descartes' Rule of Signs Tell You?
Descartes' rule of sign tells us that a polynomial f(x) cannot have more positive real zeros than the number of sign changes in it and f(x) cannot have more number of negative real zeros than the number of sign changes in f(-x).
How to Construct Descartes' Rule of Signs Chart?
Descartes' rule of signs chart has three columns: number of positive real zeros, number of negative real zeros, number of imaginary zeros. The first row of this table is filled by using the number of positive and negative real roots by counting the sign changes of f(x) and f(-x) respectively. Every other row entries can be obtained by reducing the first row values by even numbers. The number of imaginary zeros of each row is obtained by subtracting the sum of the corresponding number of positive and negative real zeros from the degree of the polynomial.
How do you Use Descartes's Rule of Signs to Find Complex Roots?
Using the rule of signs, we can find the number of positive + negative real roots. By subtracting this sum from the degree of the polynomial, we can find the possible number of complex roots.
Can we Find Zeros Using Descartes' Rule of Signs?
No, we cannot find the zeros of a polynomial using Descartes' rule of signs. We can only determine the maximum number of positive real zeros and negative real zeros by this rule.
What is the Meaning of Sign Change in Descartes' Rule of Signs?
A sign change in Descartes' rule of sign refers to a sign change from + to - or from - to +. For example, the number of sign changes in f(x) = x2 - 2x + 1 is 2.