Question

Which polynomial is prime? • x^2+7 • x^2-25 • 3x^2-27 • 2x^2-8

Answer 1

The prime polynomial is x^2 - 25. Option B is answer.

Step-by-step explanation:

A prime polynomial is a polynomial that cannot be further factored into non-constant and non-trivial polynomials with integer coefficients. Let's analyze each option:

x^2 + 7: This can be factored as (x + √7)(x - √7), making it not prime.

x^2 - 25: This can be factored as (x + 5)(x - 5), but since 5 is a constant integer, it still qualifies as a prime polynomial.

3x^2 - 27: This can be factored as 3(x^2 - 9), which further factors into 3(x + 3)(x - 3). Since 3 is a constant integer, it remains a prime polynomial.

2x^2 - 8: This can be factored as 2(x^2 - 4), which further factors into 2(x + 2)(x - 2). Again, 2 is a constant integer, making it a prime polynomial.

Therefore, the only option that cannot be factored further into non-constant and non-trivial polynomials is x^2 - 25. It remains prime regardless of the constant factor of -1.

Option B is answer.

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Complete Question

Which polynomial is prime?

x^2+7

x^2-25

3x^2-27

2x^2-8

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Answer 2

Explanation

Polynomials can be factored using their zeros and leading coefficient like this:

`f(x)=a(x-r_1)(x-r_2)...(x-r_n)`

Where a is the leading coefficient and the r's are the zeros of the