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Answer :
To determine which expression is a prime polynomial, we need to check if each polynomial is irreducible over the integers. A prime polynomial is one that cannot be factored into the product of two non-constant polynomials with integer coefficients.
Let's explore each expression:
A. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored as follows:
- The greatest common factor (GCF) of [tex]\(3x^2\)[/tex] and [tex]\(18y\)[/tex] is 3, so we can factor it out:
[tex]\[
3x^2 + 18y = 3(x^2 + 6y)
\][/tex]
Since it can be factored, it is not a prime polynomial.
B. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
Check if there is a common factor:
- The expression has a common factor of [tex]\(x\)[/tex]:
[tex]\[
10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(x^3 - 27y^6\)[/tex]
This expression can be expressed as a difference of cubes:
- It fits the formula [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex].
[tex]\[
x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
Since it can be factored, it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Check for factorization:
- This polynomial does not factor easily. It appears to be irreducible over the integers.
Since option D cannot be factored further with integer coefficients, it is considered a prime polynomial.
Therefore, the correct answer is:
D. [tex]\(x^4 + 20x^2 - 100\)[/tex] is a prime polynomial.
Let's explore each expression:
A. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored as follows:
- The greatest common factor (GCF) of [tex]\(3x^2\)[/tex] and [tex]\(18y\)[/tex] is 3, so we can factor it out:
[tex]\[
3x^2 + 18y = 3(x^2 + 6y)
\][/tex]
Since it can be factored, it is not a prime polynomial.
B. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
Check if there is a common factor:
- The expression has a common factor of [tex]\(x\)[/tex]:
[tex]\[
10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(x^3 - 27y^6\)[/tex]
This expression can be expressed as a difference of cubes:
- It fits the formula [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex].
[tex]\[
x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
Since it can be factored, it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Check for factorization:
- This polynomial does not factor easily. It appears to be irreducible over the integers.
Since option D cannot be factored further with integer coefficients, it is considered a prime polynomial.
Therefore, the correct answer is:
D. [tex]\(x^4 + 20x^2 - 100\)[/tex] is a prime polynomial.
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