Which polynomial is prime?

[tex]7 x^2-35 x+2 x-10[/tex]

[tex]9 x^3+11 x^2+3 x-33[/tex]

[tex]10 x^3-15 x^2+8 x-12[/tex]

[tex]12 x^4+42 x^2+4 x^2+14[/tex]

After factoring each polynomial, the only one that cannot be factored further into non-constant polynomials is 9x^3 + 11x^2 + 3x - 33. Therefore, this polynomial is considered prime. The final answer is 9 x 3 + 11 x 2 + 3 x − 33 ​ .
;

Answered by Anonymous | 2025-07-04

Factor each of the given polynomials.
7 x 2 − 35 x + 2 x − 10 = ( 7 x + 2 ) ( x − 5 ) .
10 x 3 − 15 x 2 + 8 x − 12 = ( 5 x 2 + 4 ) ( 2 x − 3 ) .
12 x 4 + 42 x 2 + 4 x 2 + 14 = 2 ( 2 x 2 + 7 ) ( 3 x 2 + 1 ) .
The only polynomial that appears to be prime is 9 x 3 + 11 x 2 + 3 x − 33 , so the final answer is 9 x 3 + 11 x 2 + 3 x − 33 ​ .

Explanation

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

Factoring Polynomial 1 Let's factor the first polynomial: 7 x 2 − 35 x + 2 x − 10 . We can factor by grouping: 7 x 2 − 35 x + 2 x − 10 = 7 x ( x − 5 ) + 2 ( x − 5 ) = ( 7 x + 2 ) ( x − 5 ) Since it can be factored into non-constant polynomials, it is not prime.

Analyzing Polynomial 2 Now, let's consider the second polynomial: 9 x 3 + 11 x 2 + 3 x − 33 . From the tool, we see that it cannot be factored further. Thus, it is a candidate for a prime polynomial.

Factoring Polynomial 3 Next, we factor the third polynomial: 10 x 3 − 15 x 2 + 8 x − 12 . We can factor by grouping: 10 x 3 − 15 x 2 + 8 x − 12 = 5 x 2 ( 2 x − 3 ) + 4 ( 2 x − 3 ) = ( 5 x 2 + 4 ) ( 2 x − 3 ) Since it can be factored into non-constant polynomials, it is not prime.

Factoring Polynomial 4 Finally, we factor the fourth polynomial: 12 x 4 + 42 x 2 + 4 x 2 + 14 = 12 x 4 + 46 x 2 + 14 . We can factor out a 2: 2 ( 6 x 4 + 23 x 2 + 7 ) . Now we can factor the quadratic in x 2 : 2 ( 6 x 4 + 2 x 2 + 21 x 2 + 7 ) = 2 ( 2 x 2 + 7 ) ( 3 x 2 + 1 ) .
Since it can be factored into non-constant polynomials, it is not prime.

Conclusion Therefore, the only polynomial that is prime is 9 x 3 + 11 x 2 + 3 x − 33 .


Examples
Prime polynomials are similar to prime numbers; they can't be broken down into simpler polynomial factors, just like prime numbers can't be divided by smaller integers. In cryptography, prime numbers play a crucial role in creating secure encryption keys. Similarly, prime polynomials can be used in coding theory and other areas of mathematics and computer science where irreducible polynomials are needed. For example, they can be used in error-correcting codes to detect and correct errors in data transmission.

Answered by GinnyAnswer | 2025-07-04