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What is Meant by Irreducible (Prime) Polynomials?
In mathematics, An irreducible polynomial or prime polynomial is defined as a polynomial with integer coefficients that cannot be factored into polynomials of lower degree, also with integer coefficients.
For example, x^3+x^5+1+x^3+x^3+x is a prime polynomial. The polynomial factors of x^3+x^5+1+x^3+x^3+x are 1 and x^3+x^5+1+x^3+x^3+x.
Here, you can't find any of the paths to obtain two integers b and c such that their product is 1 and their sum is also 1, therefore we cannot factor into linear terms (x+b)(x+c).
How do you Determine a Prime Polynomial Manually?
The simple steps that need to be followed to determine an irreducible polynomial are listed below:
- At first, look at the given polynomial and find factors with the help of Factoring Multi Variable Polynomials Calculator or else use the GCF method or factoring for the polynomial.
- In case the factors of the polynomial have 1 or itself factor then it is called a prime polynomial. Also, if the equation is factored into polynomials of a lower degree then it is said to be an irreducible or prime polynomial.
- Else, it is not a prime or irreducible polynomial.
Check out the provided solved example and understand how polynomials are determined prime or not and gain more conceptual knowledge on the polynomials. Also, verify your answers for the complex polynomials from our handy Determining if Polynomial is Prime Calculator in microseconds.
Consider the following polynomials and determine whether prime or not:
(i) x^5 + x^3 + x^3 + x^3 + x + 1
(ii) x^2 - 25
(i) Given polynomial is x^5 + x^3 + x^3 + x^3 + x + 1
So x^5 + x^3 + x^3 + x^3 + x + 1 = x^5 + 3 x^3 + x + 1
After simplifying x^5 + 3 x^3 + x + 1, it has two factors ie., 1 and x^5 + 3 x^3 + x + 1
Therefore, x^5 + x^3 + x^3 + x^3 + x + 1 is a prime polynomial.
(ii) Given polynomial is x^2 - 25
The factors of x^2 - 25 are (x + 5) (x - 5) after applying the GCF of a polynomial.
Now, it has 4 factors such as 1,x + 5,x - 5,x^2 - 25
Hence, x^2 - 25 is not a Prime or irreducible polynomial.
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FAQs on Determining if Polynomial is Prime Calculator
1. How do you determine if a polynomial is prime?
You easily determine if a polynomial is prime or not by using our handy online determine if Polynomial is Prime calculator.
2. How do you know if a factor is prime?
The factor of the number or polynomial is divided only by 1 or itself is known as prime number or polynomial.
3. What is an example of a prime polynomial?
x^2+8x-5 is a Prime Polynomial example where it has two factors ie., 1 and itself.
4. Is x^3+3x^2-2x-6 a prime polynomial?
After simplification the given polynimial is factored into x^3 + 3 x^2 + (-2) x - 6 = (x + 3) (x^2 - 2).It has more than 2 factors like 1, x + 3, x^2 - 2, x^3 + 3 x^2 + (-2) x - 6 so it is not a prime polynomial.