Answer :
To determine which of the given expressions is a prime polynomial, we need to understand what a prime polynomial is. A prime polynomial (or an irreducible polynomial) is a polynomial that cannot be factored into the product of two non-constant polynomials.
Let's examine each option:
A. [tex]$10x^4 - 5x^3 + 70x^2 + 3x$[/tex]
This polynomial can likely be factored since:
- It has a common factor for all terms. In this case, we can initially factor out an `x`, giving us:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- But notice this doesn't complete the factorization as the cubic inside may factor further. It's quite possible, but difficult to do by inspection. For now, this is not obviously a prime polynomial.
B. [tex]$x^4 + 20x^2 - 100$[/tex]
Let's try factoring:
- This polynomial looks like it might factor into quadratics. Let's set:
[tex]\[
(x^2 + a)(x^2 + b)
\][/tex]
- Expanding and setting terms, you can solve to try matching terms with the expression, but it’s complex and often doesn't succeed without evident factors. So, it might be irreducible at first glance without obvious factorization routes.
C. [tex]$x^3 - 27y^6$[/tex]
This is a difference of cubes:
- The expression is of the form [tex]\(a^3 - b^3\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex].
- Difference of cubes factors as:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
- Substituting [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
(x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
Thus, it can be factored and is not a prime polynomial.
D. [tex]$3x^2 + 18y$[/tex]
This polynomial can be factored because:
- There is a common factor of 3:
[tex]\[
3(x^2 + 6y)
\][/tex]
Again, it has been factored and is not a prime polynomial.
Thus, after checking thoroughly, option B, [tex]$x^4 + 20x^2 - 100$[/tex], is the one that's harder to factor definitively (though not proven here, only suggested) and typically tends to remain unfactorable by simple inspection compared to others, making it a candidate for a prime polynomial without evident simple factors.
Correct Answer: B. [tex]$x^4 + 20x^2 - 100$[/tex] (candidate for prime polynomial)
Note: Always make sure to get a polynomial tested further using reliable factorization or algebra tools for a final certainty in different contexts if doing this in an analytical setting.
Let's examine each option:
A. [tex]$10x^4 - 5x^3 + 70x^2 + 3x$[/tex]
This polynomial can likely be factored since:
- It has a common factor for all terms. In this case, we can initially factor out an `x`, giving us:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- But notice this doesn't complete the factorization as the cubic inside may factor further. It's quite possible, but difficult to do by inspection. For now, this is not obviously a prime polynomial.
B. [tex]$x^4 + 20x^2 - 100$[/tex]
Let's try factoring:
- This polynomial looks like it might factor into quadratics. Let's set:
[tex]\[
(x^2 + a)(x^2 + b)
\][/tex]
- Expanding and setting terms, you can solve to try matching terms with the expression, but it’s complex and often doesn't succeed without evident factors. So, it might be irreducible at first glance without obvious factorization routes.
C. [tex]$x^3 - 27y^6$[/tex]
This is a difference of cubes:
- The expression is of the form [tex]\(a^3 - b^3\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex].
- Difference of cubes factors as:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
- Substituting [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
(x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
Thus, it can be factored and is not a prime polynomial.
D. [tex]$3x^2 + 18y$[/tex]
This polynomial can be factored because:
- There is a common factor of 3:
[tex]\[
3(x^2 + 6y)
\][/tex]
Again, it has been factored and is not a prime polynomial.
Thus, after checking thoroughly, option B, [tex]$x^4 + 20x^2 - 100$[/tex], is the one that's harder to factor definitively (though not proven here, only suggested) and typically tends to remain unfactorable by simple inspection compared to others, making it a candidate for a prime polynomial without evident simple factors.
Correct Answer: B. [tex]$x^4 + 20x^2 - 100$[/tex] (candidate for prime polynomial)
Note: Always make sure to get a polynomial tested further using reliable factorization or algebra tools for a final certainty in different contexts if doing this in an analytical setting.
