Next I am going to explain how we can take advantage of the properties of the roots of a polynomial.
The roots of a polynomial have many properties that will help us to solve exercises with polynomials, especially to factorize and simplify the polynomials.
Now we go to the root properties of a polynomial :).
Roots properties of a polynomial
We are going to indicate each of the root properties of a polynomial and how we can apply them:
1 – The roots of the polynomial are also divisors of its independent term.
For example, in the polynomial:
The roots or solutions are: x = 2 and x = -4. Both 2 and -4 are divisors of 8
This property can be used to check if the roots obtained are correct.
2 – Each root of the polynomial, x = a, can be written in the form of a binomial (x-a)
If we notice, if for example in solution x = 2, we place everything in the first member, we have:
x = 2 ⇒ x-2 = 0
That’s why solutions can become that binomial.
Following the example of the previous polynomial, each solution will correspond to a binomial:
- x = 2 ⇒ (x-2)
- x = -4 ⇒ (x-(-4) = (x+4)
A polynomial has as many form (x-a) binomials as it does solutions
This property alone serves to explain the root properties of a polynomial that follow.
3 – A polynomial can be written as the product of all its binomials (x-a):
If we solve the polynomial in the form of binomials, we will see that it has the same solutions, so the polynomial is equivalent:
x-2 = 0 ⇒ x = 2
x+4 = 0 ⇒ x = -4
This is the most important property since factoring polynomials, by being able to express them as a product of binomials.
4 – The sum of the degrees of all binomials is equal to the degree of the original polynomial.
Our example polynomial is grade 2:
It can be broken down into two binomials: (x-2) and (x+4), which are both grade 1, which together add up to the grade of the original polynomial
5 – If one of the roots of the polynomial is x = 0, the corresponding factor is x, since its binomial will be x-0, i.e. x.
This occurs when the polynomial does not have an independent term and therefore we can draw a common factor from the x.
Property also used when factoring.
6 – When a polynomial cannot be expressed in factors or binomials, it is called irreducible or primo.
We are presented with this case when the polynomial has no real solutions, e.g.:
This polynomial could not be expressed in any other way, being irreducible.