Select the correct answer.
Which expression is a prime polynomial?
A. $10 x^4-5 x^3+70 x^2+3 x$
B. $x^3-27 y^6$
C. $3 x^2+18 y$
D. $x^4+20 x^2-100$
Answer (1)
Factor each polynomial to see if it can be expressed as a product of simpler polynomials.
Option A, B, and C can be factored, so they are not prime.
Option D, x 4 + 20 x 2 − 100 , cannot be factored easily with integer coefficients, making it a prime polynomial.
Therefore, the correct answer is x 4 + 20 x 2 − 100 .
Explanation
Understanding Prime Polynomials A prime polynomial is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. We need to check each option to see if it can be factored to determine which one is prime.
Checking Each Option Let's examine each option:
Option A: 10 x 4 − 5 x 3 + 70 x 2 + 3 x . We can factor out an x from each term: x ( 10 x 3 − 5 x 2 + 70 x + 3 ) . Since it can be factored, it is not a prime polynomial.
Option B: x 3 − 27 y 6 . This can be recognized as a difference of cubes, where x 3 − ( 3 y 2 ) 3 . Using the difference of cubes factorization, a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) , we have ( x − 3 y 2 ) ( x 2 + 3 x y 2 + 9 y 4 ) . Since it can be factored, it is not a prime polynomial.
Option C: 3 x 2 + 18 y . We can factor out a 3 from each term: 3 ( x 2 + 6 y ) . Since it can be factored, it is not a prime polynomial.
Option D: x 4 + 20 x 2 − 100 . Let u = x 2 . Then the expression becomes u 2 + 20 u − 100 . We can try to find two numbers that multiply to -100 and add to 20. However, there are no such integer factors. Using the quadratic formula to solve for the roots of u 2 + 20 u − 100 = 0 , we have:
u = 2 ( 1 ) − 20 ± 2 0 2 − 4 ( 1 ) ( − 100 ) = 2 − 20 ± 400 + 400 = 2 − 20 ± 800 = 2 − 20 ± 20 2 = − 10 ± 10 2
So, x 2 = − 10 + 10 2 or x 2 = − 10 − 10 2 . This means x = ± − 10 + 10 2 or x = ± − 10 − 10 2 . Since the roots are not rational, we cannot easily factor this polynomial over the integers. The factored form involves irrational coefficients, and thus, we consider this polynomial to be prime.
Conclusion Based on the analysis, the polynomial that cannot be factored into simpler polynomials with integer coefficients is x 4 + 20 x 2 − 100 .
Examples
Prime polynomials are similar to prime numbers in that they cannot be broken down into smaller factors. In cryptography, prime polynomials can be used in the construction of finite fields, which are essential for secure communication and data encryption. For example, in the Advanced Encryption Standard (AES), finite fields based on prime polynomials are used to perform mathematical operations that scramble and obscure data, making it unreadable to unauthorized parties. Understanding prime polynomials helps in designing and analyzing cryptographic algorithms.
