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Which expression is a prime polynomial?

A. [tex]x^4 + 20x^2 - 100[/tex]
B. [tex]3x^2 + 18y[/tex]
C. [tex]x^3 - 2x^2[/tex]
D. [tex]10x^4 - 5x^3 + 70x^2 + 3x[/tex]

Answer :

To determine which expression is a prime polynomial, we need to check each of the given polynomials to see if they can be factored. A prime polynomial cannot be factored into two or more non-trivial polynomials with coefficients in the same set (usually integers or rational numbers).

Let's evaluate each option:

A. [tex]\(x^4 + 20x^2 - 100\)[/tex]:

This polynomial might look complex, but let's check if it can be factored. We can try substitution by setting [tex]\( u = x^2 \)[/tex] to transform it to a quadratic form:

[tex]\[ u^2 + 20u - 100 \][/tex]

Now, check if this quadratic can be factored. A basic approach to factor quadratics is to look for two numbers that multiply to [tex]\(-100\)[/tex] and add up to [tex]\(20\)[/tex]. However, such numbers are not readily visible, suggesting that the quadratic isn't easily factorizable over the integers. Further methods or use of the quadratic formula may confirm irreducibility.

B. [tex]\(3x^2 + 18y\)[/tex]:

This expression can be factored by taking out the greatest common factor:

[tex]\[ 3x^2 + 18y = 3(x^2 + 6y) \][/tex]

Since it can be factored, it is not a prime polynomial.

C. [tex]\(x^3 - 2x^2\)[/tex]:

Again, look for the greatest common factor:

[tex]\[ x^3 - 2x^2 = x^2(x - 2) \][/tex]

This expression can be factored, indicating it is not prime.

D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]:

Here, notice the common factor in the terms:

[tex]\[ 10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3) \][/tex]

Since it can be factored, this polynomial is not prime.

Among the options, [tex]\(x^4 + 20x^2 - 100\)[/tex] stands out as a polynomial that couldn't easily be factored through basic methods. Based on this analysis, option A is likely the expression that represents a prime polynomial.

The correct answer is A.

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