Answer :
To determine which expression is a prime polynomial, we need to check if each polynomial can be factored further. A prime polynomial is one that cannot be factored into simpler polynomials with integer coefficients.
Let's analyze each option:
A. [tex]\(3x^2 + 18y\)[/tex]
- This polynomial can be factored by taking out the greatest common factor (GCF), which is 3:
[tex]\[
3(x^2 + 6y)
\][/tex]
- Since it can be factored, it is not a prime polynomial.
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression can also be factored further using substitution. Let [tex]\(u = x^2\)[/tex], then it becomes a quadratic:
[tex]\[
u^2 + 20u - 100
\][/tex]
- We can use the quadratic formula or factor by grouping to see if it can be broken down into factors with integer coefficients, finding the GCF or checking for special patterns. It turns out it can be factored, so it's not a prime polynomial.
C. [tex]\(x^3 - 27y^6\)[/tex]
- This can be expressed using the difference of cubes 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]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Here, we see that we can factor out an x:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- Further examination may show additional factorizations, indicating that it is not prime.
Since all the given polynomials can be factored into simpler parts, they are not considered prime polynomials. As a result, none of the options given is a prime polynomial.
Let's analyze each option:
A. [tex]\(3x^2 + 18y\)[/tex]
- This polynomial can be factored by taking out the greatest common factor (GCF), which is 3:
[tex]\[
3(x^2 + 6y)
\][/tex]
- Since it can be factored, it is not a prime polynomial.
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression can also be factored further using substitution. Let [tex]\(u = x^2\)[/tex], then it becomes a quadratic:
[tex]\[
u^2 + 20u - 100
\][/tex]
- We can use the quadratic formula or factor by grouping to see if it can be broken down into factors with integer coefficients, finding the GCF or checking for special patterns. It turns out it can be factored, so it's not a prime polynomial.
C. [tex]\(x^3 - 27y^6\)[/tex]
- This can be expressed using the difference of cubes 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]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Here, we see that we can factor out an x:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- Further examination may show additional factorizations, indicating that it is not prime.
Since all the given polynomials can be factored into simpler parts, they are not considered prime polynomials. As a result, none of the options given is a prime polynomial.
