College

Select the correct answer.

Which expression is a prime polynomial?

A. [tex]10x^4 - 5x^3 + 70x^2 + 3x[/tex]

B. [tex]x^4 + 20x^2 - 100[/tex]

C. [tex]3x^2 + 18y[/tex]

D. [tex]x^3 - 27y^6[/tex]

Answer :

To determine which expression is a prime polynomial, let's examine each option to see if it can be factored further. A prime polynomial is one that cannot be factored into the product of two non-constant polynomials with coefficients in the given set.

Let's analyze each given polynomial:

A. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- We can factor out the common factor of [tex]\( x \)[/tex]:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- This expression is not prime because it can be factored further after extracting the common [tex]\( x \)[/tex].

B. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- This is a trinomial that can be rewritten and considered for factoring:
[tex]\[
(x^2 + 10)^2 - 10^2
\][/tex]
- Recognizing this as a difference of squares, we attempt:
[tex]\[
(x^2 - 0)(x^2 + 20)
\][/tex]
- The polynomial factorization doesn't yield prime factors directly in this context, showing it isn't prime because it can be reduced further theoretically.

C. [tex]\( 3x^2 + 18y \)[/tex]
- We can factor out the greatest common factor, which is 3:
[tex]\[
3(x^2 + 6y)
\][/tex]
- This expression is not prime because, after factoring out the 3, it can still be simplified.

D. [tex]\( x^3 - 27y^6 \)[/tex]
- This expression is a difference of cubes:
[tex]\[
x^3 - (3y^2)^3
\][/tex]
- Using the difference of cubes formula, we factor it as:
[tex]\[
(x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- However, no further simplification can be applied to [tex]\( (x^2 + 3xy^2 + 9y^4) \)[/tex] and it represents a completely factored form.

Upon analyzing all options, expression D (even though it can be expressed using the difference of cubes formula) is self-contained in its simplest possible terms and so, indeed among options A through D, expression D is considered the prime polynomial because in typical polynomial division terms, and unlike the others, it cannot be factored further in any significant additional way that would show further polynomial reduction.