The law of signs, more formally known as Descartes’ Rule of Signs, is a useful theorem in algebra for determining the number of positive and negative real roots of a polynomial. This rule provides a way to predict how many positive or negative solutions a polynomial equation might have based on the signs of its coefficients.
How It Works
Positive Roots
To find the number of positive real roots of a polynomial, follow these steps:
- Arrange the polynomial in standard form, with terms ordered from highest to lowest degree.
- Count the number of sign changes between consecutive non-zero coefficients.
The number of positive real roots is either equal to the number of sign changes or less than that by an even number. For example, if there are 3 sign changes, the polynomial could have 3, 1, or 0 positive real roots.
Negative Roots
To find the number of negative real roots, substitute $-x$ for $x$ in the polynomial and then follow the same steps as above to count the sign changes. The number of negative real roots will also be equal to the number of sign changes or less than that by an even number.
Example
Consider the polynomial $P(x) = x^4 – 3x^3 + 2x^2 – x + 5$
- Positive Roots: The polynomial in standard form is already given. The coefficients are
$1, -3, 2, -1, 5$. The sign changes are:- $1$ to $-3$ (change)
- $-3$ to $2$ (change)
- $2$ to $-1$ (change)
- $-1$ to $5$ (change)
There are 4 sign changes, so there could be 4, 2, or 0 positive real roots.
- Negative Roots: Substitute $-x$ for $x$ in the polynomial to get $P(-x) = (-x)^4 – 3(-x)^3 + 2(-x)^2 – (-x) + 5 = x^4 + 3x^3 + 2x^2 + x + 5$. The coefficients are $1, 3, 2, 1, 5$. There are no sign changes, so there are no negative real roots.
Conclusion
Descartes’ Rule of Signs is a handy tool for predicting the number of positive and negative real roots of a polynomial. While it doesn’t give the exact number of roots, it provides a range of possibilities, making it a valuable step in solving polynomial equations.