Which polynomial is prime?

$x^2+3 x^2+2 x+6$
$x^3+3 x^2-2 x-6$
$10 x^2 4 x+3 x+6$
$10 x^2-10 x+6 x-6$

Simplify and attempt to factor each polynomial.
4 x 2 + 2 x + 6 = 2 ( 2 x 2 + x + 3 ) is not prime.
x 3 + 3 x 2 − 2 x − 6 = ( x 2 − 2 ) ( x + 3 ) is not prime.
10 x 2 + 7 x + 6 is prime.
10 x 2 − 4 x − 6 = 2 ( 5 x + 3 ) ( x − 1 ) is not prime.
The prime polynomial is 10 x 2 + 7 x + 6 ​ .

Explanation

Problem Analysis We are given four polynomials and asked to determine which one is prime. A prime polynomial is one that cannot be factored into non-constant polynomials. We will analyze each polynomial to determine if it can be factored.

Listing Polynomials The given polynomials are:

x 2 + 3 x 2 + 2 x + 6

x 3 + 3 x 2 − 2 x − 6

10 x 2 + 4 x + 3 x + 6

10 x 2 − 10 x + 6 x − 6

Factoring Polynomials Let's simplify and factor each polynomial:

x 2 + 3 x 2 + 2 x + 6 = 4 x 2 + 2 x + 6 = 2 ( 2 x 2 + x + 3 ) . This polynomial can be factored, so it is not prime.

x 3 + 3 x 2 − 2 x − 6 = x 2 ( x + 3 ) − 2 ( x + 3 ) = ( x 2 − 2 ) ( x + 3 ) . This polynomial can be factored, so it is not prime.

10 x 2 + 4 x + 3 x + 6 = 10 x 2 + 7 x + 6 . From the tool, we know that this polynomial cannot be factored further.

10 x 2 − 10 x + 6 x − 6 = 10 x 2 − 4 x − 6 = 2 ( 5 x 2 − 2 x − 3 ) = 2 ( 5 x + 3 ) ( x − 1 ) . This polynomial can be factored, so it is not prime.

Identifying Prime Polynomial Based on the analysis, the polynomial 10 x 2 + 7 x + 6 cannot be factored into non-constant polynomials.


Examples
Prime polynomials are analogous to prime numbers in that they cannot be broken down into simpler factors. This concept is crucial in cryptography, where prime polynomials are used to construct finite fields, which are essential for secure data transmission and encryption algorithms. Understanding prime polynomials helps in designing robust cryptographic systems that protect sensitive information from unauthorized access.

Answered by GinnyAnswer | 2025-07-08