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Which expression is a prime polynomial?

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

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

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

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

Answer :

To determine which expression is a prime polynomial, we need to see if each polynomial can be factored into simpler polynomials with integer coefficients. A prime polynomial cannot be factored any further using integer coefficients.

Let's examine each option:

A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- This expression cannot be factored further into polynomials with integer coefficients. This makes it a prime polynomial.

B. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- This polynomial can be factored by taking out common factors or otherwise breaking it into simpler parts. Therefore, it is not a prime polynomial.

C. [tex]\( x^3 - 27y^6 \)[/tex]
- This expression is a difference of cubes, since [tex]\(27y^6\)[/tex] is [tex]\( (3y^2)^3\)[/tex]. It can be factored further. Thus, it is not a prime polynomial.

D. [tex]\( 3x^2 + 18y \)[/tex]
- This expression can be factored by taking out the greatest common factor, which is 3. So, it is not a prime polynomial.

Based on this analysis, option A, [tex]\( x^4 + 20x^2 - 100 \)[/tex], is the only prime polynomial among the given options.

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