Root properties and factors of a polynomial.

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:properties of the roots and factors of a 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):

properties of the roots of a polynomial

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)(x+4)=0

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:

properties and roots of polynomials

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.:

properties of polynomials

This polynomial could not be expressed in any other way, being irreducible.