On this post you will find what the roots (or zeros) of a polynomial are and how to calculate all the roots of a polynomial. Also, you will see examples and exercises solved step by step of polynomial roots.
Table of Contents
What are the roots of a polynomial?
The roots (or zeros) of a polynomial are the values of x for which the polynomial is equal to zero, that is, x=a is a polynomial root if P(a)=0.
For example, let P(x) be a polynomial:
We can check that one of the roots of the polynomial is x=1 evaluating the polynomial at that point:
On the other hand, 3 is not a root of this polynomial because it is not a value that cancels the polynomial, in other words, if we substitute x for 3 the polynomial is nonzero:
I hope that now you understand better what the root of a polynomial is, but wouldn’t you like to know how many roots a polynomial has? Or how to find all the roots of a polynomial? Well, this is precisely what we are going to see in the next section.
How to find all the roots of a polynomial
To find all the roots of a polynomial, you must do the following steps:
- First, find all the divisors (or factors) of the constant term of the polynomial.
- Second, evaluate the polynomial at all the values found in the previous step.
- Third, if the evaluation of a number results in zero, this number is a root of the polynomial. Otherwise, that number is not a root of the polynomial.
Example
Next we are going to solve an example step by step so that you can better understand how to find the zeros of a polynomial.
- What are all the roots (or zeros) of the following polynomial?
First of all, we must find the factors of the constant term of the polynomial, because any root of a polynomial is also a factor of the constant term. So, the factors of 6 are:
Factors of 6: +1, -1, +2, -2, +3, -3
Remember that if a number is a factor, its negative is also. Since a number is divisible by positive and negative numbers.
So the possible roots or zeros of the polynomial are: ±1, ±2, ±3. Therefore, we must evaluate the polynomial at all these values. For that, we substitute all these values in the expression of the polynomial for x:
So the polynomial equals to zero only when the variable x is +2 or +3, so these are the roots of the polynomial:
Roots or zeros of the polynomial: +2 and +3
On the other hand, note that the polynomial has as many roots as its degree. In the section of the properties of the roots of a polynomial (below), we will see why this characteristic always holds for any polynomial.
Properties of polynomial roots
The polynomial roots (or zeros) have the following characteristics:
- As we have seen before, the integer roots (or zeros) of a polynomial are divisors of the constant term of the polynomial.
- If we know all the roots of a polynomial, we can express the polynomial in the form of products of binomials of the type
For example, the polynomial 
has three roots, which are 
, and 
Thus, we can rewrite the polynomial in the form of three multiplications of factors, each one formed by the variable 
and a root changed sign:
- A polynomial has as many complex roots as its degree indicates. So a second-degree polynomial will have 2 roots, a third-degree polynomial will have 3 roots, a fourth-degree polynomial will have 4 roots, and so on.
- If a polynomial does not have a constant term, it means that at least one of its roots is 0. Then, the rest of the polynomial roots are divisors of the coefficient of the lowest degree monomial.
For example, the following polynomial has no constant term:
So a root of the polynomial must necessarily be 0. And the rest of the roots are factors of the coefficient of the term of lowest degree, that is, -2. In particular, the other polynomial zeros are 
and 
Roots or zeros of the polynomial: 0, +1 and -2
- When the roots of a polynomial cannot be determined, we say that it is an irreducible polynomial.
For example, we are going to try to calculate the roots of the following polynomial:
The only possible roots of the polynomial are the factors of -1, that is, -1 and +1. So we evaluate the polynomial at these values:
In no case does the polynomial result in 0, therefore it does not have integer roots and it is an irreducible polynomial over the integers.
- When the polynomial is composed of the product of several polynomials, it is not necessary to compute the product to calculate the roots, but the roots of the polynomial are the roots of each factor.
As an example, if we have the following polynomial:
From the second property of the polynomial roots, we can deduce that the root of the polynomial on the left is +2 and the root of the polynomial on the right is -1.
Therefore, the roots of the polynomial resulting from the multiplication of the two factors are their respective roots, that is, +2 and -1.
Practice problems on finding polynomial roots
Problem 1
Determine whether 
is a root of the following polynomial:
See solution
To know if x=-4 is a root of the polynomial we must evaluate it at that value. So:
The polynomial evaluated at is null, so it is a root of the polynomial.
Problem 2
Find all the roots of the following quadratic polynomial:
See solution
First, to find the possible roots of the polynomial we have to find the divisors of the constant term. Thus, the divisors of 2 are:
Divisors of 2: +1, -1, +2, -2
So the possible polynomial roots or zeros are ±1 and ± 2. Therefore, we must evaluate the polynomial at all these values:
So the polynomial is zero when x is +1 or +2, so these are the polynomial roots:
Roots or zeros of the polynomial: +1 and +2
Problem 3
Find the roots of the following cubic polynomial:
See solution
First we must find all the factors of the constant term, since the root of a polynomial is also a factor of its constant term. The factors of 4 are:
Divisors of 4: +1, -1, +2, -2, +4, -4
So the possible polynomial roots or zeros are ±1, ±2 and ±4. Now we have to evaluate the polynomial at all these values:
So the polynomial roots are:
Roots or zeros of the polynomial: +1, +2 and -2
Problem 4
Find all the zeros of the following polynomial:
See solution
In this case, the polynomial has no constant term. Therefore, according to the fourth property of the polynomial roots explained above, we know that one of the roots of the polynomial must be 0.
Roots of the polynomial: 
Moreover, in this case the possible roots are not the factors of the constant term, but of the coefficient of the lowest degree term, that is 8:
Divisors of 8: +1, -1, +2, -2, +4, -4, +8, -8
So the possible roots or zeros of the polynomial are ±1, ±2, ±4 and ±8.
So the polynomial equals to zero if x is +2 or +4, so these values are roots of the polynomial. However, we also have to add the root 0 that we found at the beginning of the problem. In conclusion, all the roots of the polynomial are:
Roots or zeros of the polynomial: 0, +2 and +4
Problem 5
Use the properties of the polynomial roots to calculate all the zeros of the following polynomial:
See solution
As we have seen in the sixth property of the polynomial roots, when the polynomial is formed by the product of factors, it is not necessary to calculate all the roots, since the roots of the entire polynomial are the roots of each factor.
Furthermore, from the second property of the polynomial roots, we can deduce that the root of the first factor is +1 and the root of the second factor is -3.
Therefore, we only need to find the roots of the last factor. To do this, we find the divisors of the constant term (-2):
Divisors of -2: +1, -1, +2, -2
And evaluate the polynomial at all these values:
So the roots of the polynomial on the right are -1 and 2.
Therefore, the roots of the polynomial are all the roots found:
Roots or zeros of the polynomial : +1, -1, +2, -3
FAQs
How do you find the roots or zeros of a polynomial function? ›
We can find the Roots or Zeros of a polynomial by setting the polynomial equal to 0 and factoring. Some are easier to factor than others! If we cannot factor the polynomial, but know one of the roots, we can divide that factor into the polynomial.
How do you find all zeros of a polynomial? ›Use the Rational Zero Theorem to list all possible rational zeros of the function. Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate.
How do you know how many roots zeros a polynomial has? ›How many roots does a polynomial have? The number of roots of any polynomial is depended on the degree of that polynomial. Suppose n is the degree of a polynomial p(x), then p(x) has n number of roots. For example, if n = 2, the number of roots will be 2.
What is the easiest way to find the zeros of a function? ›In general, given the function, f(x), its zeros can be found by setting the function to zero. The values of x that represent the set equation are the zeroes of the function. To find the zeros of a function, find the values of x where f(x) = 0.
How do you find the root trick? ›- Step 1: Pair the digits from right to left, 404 01.
- Step 2: Check the unit digit of the number and compare it with the above table. ...
- Step 3: Now, the first three digits are 404. ...
- Step 4: Hence, the required square root is 201.
To solve a polynomial equation, first write it in standard form. Once it is equal to zero, factor it and then set each variable factor equal to zero. The solutions to the resulting equations are the solutions to the original.
How can you quickly determine the number of roots a polynomial? ›Look at the leading term of the equation to determine the number of roots of a polynomial. A leading term is a term with the highest power. The highest power of the equation is 1 and will have one root.
How do you find out how many roots an equation has? ›The value of the discriminant shows how many roots f(x) has: - If b2 – 4ac > 0 then the quadratic function has two distinct real roots. - If b2 – 4ac = 0 then the quadratic function has one repeated real root. - If b2 – 4ac < 0 then the quadratic function has no real roots.
What method is the fastest way on finding the roots of the polynomial? ›The simplest root-finding algorithm is the bisection method.