Answer :
To determine which expression is a prime polynomial, we first need to understand what a prime polynomial is. A polynomial is considered prime if it cannot be factored into polynomials of lower degree with integer coefficients.
Let's look at the given options one by one:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial is of degree 4. We would attempt to factor it into products of lower-degree polynomials. However, this expression cannot be factored further with integer coefficients, so it remains as it is. Hence, it's a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
This expression can be factored using the difference of cubes formula:
[tex]\[
x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
Therefore, this expression is not a prime polynomial because it can be factored.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
Notice each term contains an [tex]\(x\)[/tex], so we can factor out the common factor [tex]\(x\)[/tex]:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
Because it can be factored further, this polynomial is not prime.
D. [tex]\(3x^2 + 18y\)[/tex]
There is a common factor of 3 here:
[tex]\[
3(x^2 + 6y)
\][/tex]
Since this expression can be factored, it is not a prime polynomial either.
From the analysis, the only polynomial that cannot be factored further with integer coefficients is:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Thus, the expression that is a prime polynomial is found in option A.
Let's look at the given options one by one:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial is of degree 4. We would attempt to factor it into products of lower-degree polynomials. However, this expression cannot be factored further with integer coefficients, so it remains as it is. Hence, it's a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
This expression can be factored using the difference of cubes formula:
[tex]\[
x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
Therefore, this expression is not a prime polynomial because it can be factored.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
Notice each term contains an [tex]\(x\)[/tex], so we can factor out the common factor [tex]\(x\)[/tex]:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
Because it can be factored further, this polynomial is not prime.
D. [tex]\(3x^2 + 18y\)[/tex]
There is a common factor of 3 here:
[tex]\[
3(x^2 + 6y)
\][/tex]
Since this expression can be factored, it is not a prime polynomial either.
From the analysis, the only polynomial that cannot be factored further with integer coefficients is:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Thus, the expression that is a prime polynomial is found in option A.
