Which polynomial is prime?

[tex]3 x^3+3 x^2-2 x-2[/tex]

[tex]3 x^3-2 x^2+3 x-4[/tex]

[tex]4 x^3+2 x^2+6 x+3[/tex]

[tex]4 x^3+4 x^2-3 x-3[/tex]

None of the given polynomials are prime because each can be factored into lower-degree polynomials. Specifically, all four polynomials can be expressed as products of non-constant polynomials. Therefore, there is no prime polynomial among the options provided.
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Answered by Anonymous | 2025-07-04

Factor each polynomial by grouping.
3 x 3 + 3 x 2 − 2 x − 2 = ( 3 x 2 − 2 ) ( x + 1 ) .
3 x 3 − 2 x 2 + 3 x − 4 = ( x − 1 ) ∗ ( 3 x 2 + x + 4 ) .
4 x 3 + 2 x 2 + 6 x + 3 = ( 2 x 2 + 3 ) ( 2 x + 1 ) .
4 x 3 + 4 x 2 − 3 x − 3 = ( 4 x 2 − 3 ) ( x + 1 ) .
Since all polynomials can be factored, none are prime. Therefore, there is no prime polynomial among the given options.

Explanation

Understanding the Problem We are given four cubic polynomials and asked to identify the prime polynomial. A prime polynomial is one that cannot be factored into non-constant polynomials of lower degree.

Factoring the First Polynomial Let's analyze the first polynomial: 3 x 3 + 3 x 2 − 2 x − 2 . We can try factoring by grouping. 3 x 3 + 3 x 2 − 2 x − 2 = 3 x 2 ( x + 1 ) − 2 ( x + 1 ) = ( 3 x 2 − 2 ) ( x + 1 ) Since it can be factored, it is not prime.

Factoring the Second Polynomial Now, let's analyze the second polynomial: 3 x 3 − 2 x 2 + 3 x − 4 . We can try factoring by grouping or looking for rational roots. Factoring gives us: 3 x 3 − 2 x 2 + 3 x − 4 = ( x − 1 ) ∗ ( 3 x 2 + x + 4 ) Since it can be factored, it is not prime.

Factoring the Third Polynomial Next, let's analyze the third polynomial: 4 x 3 + 2 x 2 + 6 x + 3 . We can try factoring by grouping. 4 x 3 + 2 x 2 + 6 x + 3 = 2 x 2 ( 2 x + 1 ) + 3 ( 2 x + 1 ) = ( 2 x 2 + 3 ) ( 2 x + 1 ) Since it can be factored, it is not prime.

Factoring the Fourth Polynomial Finally, let's analyze the fourth polynomial: 4 x 3 + 4 x 2 − 3 x − 3 . We can try factoring by grouping. 4 x 3 + 4 x 2 − 3 x − 3 = 4 x 2 ( x + 1 ) − 3 ( x + 1 ) = ( 4 x 2 − 3 ) ( x + 1 ) Since it can be factored, it is not prime.

Conclusion Since all the given polynomials can be factored into non-constant polynomials of lower degree, none of them are prime.


Examples
Prime polynomials are analogous to prime numbers in that they cannot be factored into simpler polynomials. This concept is crucial in cryptography, where prime polynomials are used in constructing finite fields, which are essential for secure data transmission and encryption algorithms. 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. Understanding prime polynomials helps in designing robust and secure communication systems.

Answered by GinnyAnswer | 2025-07-04