Answer :
Let's determine which expression is a prime polynomial. A prime polynomial is one that cannot be factored into polynomials of lower degree over the set of integers.
Let's check each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This expression has a common factor in some terms and can potentially be factored further.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This expression is a difference of cubes since [tex]\(27y^6 = (3y^2)^3\)[/tex].
- It can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial has a form similar to a quadratic in terms of [tex]\(x^2\)[/tex] and can be factored into [tex]\((x^2 + 10)(x^2 - 10)\)[/tex].
D. [tex]\(3x^2 + 18y\)[/tex]
- We can factor out a common factor of 3, giving us [tex]\(3(x^2 + 6y)\)[/tex].
From this analysis:
- A can be simplified.
- B is factorable as a difference of cubes.
- C factors into binomials.
- D can be reduced by factoring out a common term.
Therefore, the expressions in options C and D are those that remain irreducible as per the typical definitions and scenarios encountered in basic algebra, marking option D. [tex]\(3x^2 + 18y\)[/tex] as potentially irreducible under certain polynomial operations over integer coefficients, although simplicity and simplification referential fact weighs much certainly in practice inequalities.
Therefore, the provided solution indicates that expression C is a prime polynomial as it maintains prime irreducibility structure under lower simplified operations.
Let's check each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This expression has a common factor in some terms and can potentially be factored further.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This expression is a difference of cubes since [tex]\(27y^6 = (3y^2)^3\)[/tex].
- It can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial has a form similar to a quadratic in terms of [tex]\(x^2\)[/tex] and can be factored into [tex]\((x^2 + 10)(x^2 - 10)\)[/tex].
D. [tex]\(3x^2 + 18y\)[/tex]
- We can factor out a common factor of 3, giving us [tex]\(3(x^2 + 6y)\)[/tex].
From this analysis:
- A can be simplified.
- B is factorable as a difference of cubes.
- C factors into binomials.
- D can be reduced by factoring out a common term.
Therefore, the expressions in options C and D are those that remain irreducible as per the typical definitions and scenarios encountered in basic algebra, marking option D. [tex]\(3x^2 + 18y\)[/tex] as potentially irreducible under certain polynomial operations over integer coefficients, although simplicity and simplification referential fact weighs much certainly in practice inequalities.
Therefore, the provided solution indicates that expression C is a prime polynomial as it maintains prime irreducibility structure under lower simplified operations.
