The roots of a polynomial will allow us to break down the polynomials into factors, which in turn will allow us to divide the polynomials more easily.
We will now explain what the roots of a polynomial are, and how to calculate them in order to take advantage of their properties.
What are the roots of a polynomial
The roots of a polynomial (also called zeros of a polynomial) are the values for which, the numerical value of the polynomial is equal to zero.
Remember that to calculate the numeric value of a polynomial, the variable of the polynomial must be replaced by a number. When this value is zero, the number will correspond to the root of the polynomial.
Let’s see it better with an example, which will help you identify the numbers that are roots of a polynomial from those that are not.
We have the following polynomial:
We are going to find the numerical value of the polynomial for when x=1. To do this, we substitute x with 1 and operate:
P(1)=-5, which is different from 0. Therefore, 1 would not be a zero or root of the polynomial P(x).
Let’s try with x=2:
P(2)=0, then 2 is a zero or root of the polynomial P(x).
Now it’s a little clearer what the roots of a polynomial are, isn’t it?
But don’t worry, we don’t have to try number by number until we meet them.
By the way, how many roots does a polynomial have? How can we calculate the roots of a polynomial in a more direct way? This is what we will see in the next section.
How to calculate the roots of a polynomial
How to find the roots of a polynomial?
When we look for the roots of a polynomial, we look for P(x)=0, therefore, if we directly equate the polynomial to 0, we will be left with an equation, whose solutions will be the roots of the polynomial.
For example, we are going to calculate the roots of the previous polynomial. To do this, we equal it to zero and proceed to solve it:
Being a second-degree equation, we have two solutions: x=2 and x=-4, which in turn are the roots of the polynomial, as we can see by substituting those numbers in the polynomial:
Therefore, to find directly the roots of a polynomial, we only have to equal it to zero and solve the equation.
And there are no more roots. The number of roots coincides with the number of solutions in the equation and, as a consequence, coincides with the degree of the polynomial or the equation:
No. of roots = No. of solutions = Grade of the equation
To solve the equations of degree equal to or greater than 3 you have to use the Ruffini’s rule.