Answer :
Certainly! Let's go through each expression step by step to determine if it is a prime polynomial. A prime polynomial is one that cannot be factored into smaller polynomials with real or complex coefficients (other than just multiplying by a constant).
### Expression A: [tex]\(x^4 + 20x^2 - 100\)[/tex]
First, we attempt to factor this expression. We'll look for patterns or use methods such as completing the square or substitution:
1. Substitute [tex]\(x^2 = u\)[/tex]:
[tex]\[
u^2 + 20u - 100
\][/tex]
2. Factor the quadratic in [tex]\(u\)[/tex]:
[tex]\[
u^2 + 20u - 100 = (u + 10\sqrt{3})(u - 10\sqrt{3})
\][/tex]
Substitute back [tex]\(u = x^2\)[/tex]:
[tex]\[
(x^2 + 10\sqrt{3})(x^2 - 10\sqrt{3})
\][/tex]
Therefore, [tex]\( x^4 + 20 x^2 - 100 \)[/tex] is not a prime polynomial since it can be factored.
### Expression B: [tex]\(3x^2 + 18y\)[/tex]
We'll try to factor the terms:
1. Factor out the greatest common factor (GCF) which is [tex]\(3\)[/tex]:
[tex]\[
3(x^2 + 6y)
\][/tex]
This shows [tex]\( 3x^2 + 18y \)[/tex] is not a prime polynomial because it can be factored.
### Expression C: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
Attempt to factor by grouping or other methods:
1. Factor out the GCF which is [tex]\(x\)[/tex]:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
Because of this factor, [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex] is not a prime polynomial since it can be factored.
### Expression D: [tex]\(x^3 - 27y^6\)[/tex]
Recognize this as a difference of cubes:
[tex]\[
x^3 - (3y^2)^3
\][/tex]
Difference of cubes formula is [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex]:
1. Let [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]
Thus, [tex]\( x^3 - 27 y^6 \)[/tex] is not a prime polynomial as it can be factored.
### Conclusion
After evaluating all given options, none of the expressions [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], and [tex]\(D\)[/tex] are prime polynomials. Thus, there is no correct option listed for a prime polynomial among the given choices.
### Expression A: [tex]\(x^4 + 20x^2 - 100\)[/tex]
First, we attempt to factor this expression. We'll look for patterns or use methods such as completing the square or substitution:
1. Substitute [tex]\(x^2 = u\)[/tex]:
[tex]\[
u^2 + 20u - 100
\][/tex]
2. Factor the quadratic in [tex]\(u\)[/tex]:
[tex]\[
u^2 + 20u - 100 = (u + 10\sqrt{3})(u - 10\sqrt{3})
\][/tex]
Substitute back [tex]\(u = x^2\)[/tex]:
[tex]\[
(x^2 + 10\sqrt{3})(x^2 - 10\sqrt{3})
\][/tex]
Therefore, [tex]\( x^4 + 20 x^2 - 100 \)[/tex] is not a prime polynomial since it can be factored.
### Expression B: [tex]\(3x^2 + 18y\)[/tex]
We'll try to factor the terms:
1. Factor out the greatest common factor (GCF) which is [tex]\(3\)[/tex]:
[tex]\[
3(x^2 + 6y)
\][/tex]
This shows [tex]\( 3x^2 + 18y \)[/tex] is not a prime polynomial because it can be factored.
### Expression C: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
Attempt to factor by grouping or other methods:
1. Factor out the GCF which is [tex]\(x\)[/tex]:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
Because of this factor, [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex] is not a prime polynomial since it can be factored.
### Expression D: [tex]\(x^3 - 27y^6\)[/tex]
Recognize this as a difference of cubes:
[tex]\[
x^3 - (3y^2)^3
\][/tex]
Difference of cubes formula is [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex]:
1. Let [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]
Thus, [tex]\( x^3 - 27 y^6 \)[/tex] is not a prime polynomial as it can be factored.
### Conclusion
After evaluating all given options, none of the expressions [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], and [tex]\(D\)[/tex] are prime polynomials. Thus, there is no correct option listed for a prime polynomial among the given choices.
