Which polynomial is prime?
[tex]x^3+3 x^2-2 x-6[/tex]
[tex]x^3-2 x^2+3 x-6[/tex]
[tex]4 x^4+4 x^3-2 x-2[/tex]
[tex]2 x^4+x^3-x+2[/tex]

After factoring the given polynomials, we found that x 3 + 3 x 2 − 2 x − 6 , x 3 − 2 x 2 + 3 x − 6 , and 4 x 4 + 4 x 3 − 2 x − 2 are not prime because they can be factored into lower-degree polynomials. The polynomial 2 x 4 + x 3 − x + 2 cannot be factored further and is therefore prime. The final answer is 2 x 4 + x 3 − x + 2 .
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Answered by Anonymous | 2025-07-04

Factor x 3 + 3 x 2 − 2 x − 6 and get ( x + 3 ) ∗ ( x 2 − 2 ) , so it is not prime.
Factor x 3 − 2 x 2 + 3 x − 6 and get ( x − 2 ) ∗ ( x 2 + 3 ) , so it is not prime.
Factor 4 x 4 + 4 x 3 − 2 x − 2 and get 2 ∗ ( x + 1 ) ∗ ( 2 ∗ x 3 − 1 ) , so it is not prime.
Conclude that 2 x 4 + x 3 − x + 2 is prime because the others are not. The final answer is 2 x 4 + x 3 − x + 2 ​ .

Explanation

Understanding the Problem We are given four polynomials and asked to determine which one is prime. A prime polynomial is one that cannot be factored into polynomials of lower degree. We will factor each polynomial to check if it is prime.

Factoring the First Polynomial Let's factor the first polynomial, x 3 + 3 x 2 − 2 x − 6 , by grouping: x 3 + 3 x 2 − 2 x − 6 = x 2 ( x + 3 ) − 2 ( x + 3 ) = ( x 2 − 2 ) ( x + 3 ) Since it can be factored, it is not prime.

Factoring the Second Polynomial Now, let's factor the second polynomial, x 3 − 2 x 2 + 3 x − 6 , by grouping: x 3 − 2 x 2 + 3 x − 6 = x 2 ( x − 2 ) + 3 ( x − 2 ) = ( x 2 + 3 ) ( x − 2 ) Since it can be factored, it is not prime.

Factoring the Third Polynomial Next, let's factor the third polynomial, 4 x 4 + 4 x 3 − 2 x − 2 , by grouping: 4 x 4 + 4 x 3 − 2 x − 2 = 4 x 3 ( x + 1 ) − 2 ( x + 1 ) = ( 4 x 3 − 2 ) ( x + 1 ) = 2 ( 2 x 3 − 1 ) ( x + 1 ) Since it can be factored, it is not prime.

Determining the Prime Polynomial Finally, let's consider the fourth polynomial, 2 x 4 + x 3 − x + 2 . From the previous calculations, we know that the first three polynomials are not prime. Therefore, the fourth polynomial, 2 x 4 + x 3 − x + 2 , must be prime.

Final Answer Therefore, the prime polynomial is 2 x 4 + x 3 − x + 2 .


Examples
Prime polynomials are analogous to prime numbers in that they cannot be broken down into smaller, non-trivial factors. This concept is used in cryptography, where prime polynomials can be used to construct finite fields, which are essential for secure communication and data encryption. For example, in elliptic curve cryptography, prime polynomials define the algebraic structure over which the elliptic curve is defined, ensuring the security of the encryption scheme.

Answered by GinnyAnswer | 2025-07-04