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Answer :
We are given four polynomials and need to determine which one is prime (i.e., it cannot be factored nontrivially into polynomials with integer coefficients).
Let's analyze each option:
1. Option A:
[tex]\[
3x^2 + 18y
\][/tex]
We can factor out a common factor of 3:
[tex]\[
3x^2 + 18y = 3\left(x^2 + 6y\right).
\][/tex]
Since it factors as a product of a constant and another expression, it is not prime.
2. Option B:
[tex]\[
x^3 - 27y^6
\][/tex]
This is a difference of cubes because [tex]\(27y^6 = \left(3y^2\right)^3\)[/tex]. A difference of cubes factors as:
[tex]\[
a^3 - b^3 = (a-b)(a^2+ab+b^2),
\][/tex]
so here we have:
[tex]\[
x^3 - (3y^2)^3 = \left(x - 3y^2\right)\left(x^2+ 3xy^2+ 9y^4\right).
\][/tex]
Therefore, the polynomial is composite.
3. Option C:
[tex]\[
x^4 + 20x^2 - 100
\][/tex]
This polynomial does not factor nontrivially into polynomials with integer coefficients. Since no factorization is possible other than the trivial one, it is a prime polynomial.
4. Option D:
[tex]\[
10x^4 - 5x^3 + 70x^2 + 3x
\][/tex]
We can factor out an [tex]\(x\)[/tex] from each term:
[tex]\[
10x^4 - 5x^3 + 70x^2 + 3x = x\left(10x^3 - 5x^2 + 70x + 3\right).
\][/tex]
The factor [tex]\(x\)[/tex] shows that this polynomial is composite.
Since only Option C, [tex]\(\displaystyle x^4 + 20x^2 - 100\)[/tex], does not factor nontrivially with integer coefficients, it is the prime polynomial.
Thus, the correct answer is Option C.
Let's analyze each option:
1. Option A:
[tex]\[
3x^2 + 18y
\][/tex]
We can factor out a common factor of 3:
[tex]\[
3x^2 + 18y = 3\left(x^2 + 6y\right).
\][/tex]
Since it factors as a product of a constant and another expression, it is not prime.
2. Option B:
[tex]\[
x^3 - 27y^6
\][/tex]
This is a difference of cubes because [tex]\(27y^6 = \left(3y^2\right)^3\)[/tex]. A difference of cubes factors as:
[tex]\[
a^3 - b^3 = (a-b)(a^2+ab+b^2),
\][/tex]
so here we have:
[tex]\[
x^3 - (3y^2)^3 = \left(x - 3y^2\right)\left(x^2+ 3xy^2+ 9y^4\right).
\][/tex]
Therefore, the polynomial is composite.
3. Option C:
[tex]\[
x^4 + 20x^2 - 100
\][/tex]
This polynomial does not factor nontrivially into polynomials with integer coefficients. Since no factorization is possible other than the trivial one, it is a prime polynomial.
4. Option D:
[tex]\[
10x^4 - 5x^3 + 70x^2 + 3x
\][/tex]
We can factor out an [tex]\(x\)[/tex] from each term:
[tex]\[
10x^4 - 5x^3 + 70x^2 + 3x = x\left(10x^3 - 5x^2 + 70x + 3\right).
\][/tex]
The factor [tex]\(x\)[/tex] shows that this polynomial is composite.
Since only Option C, [tex]\(\displaystyle x^4 + 20x^2 - 100\)[/tex], does not factor nontrivially with integer coefficients, it is the prime polynomial.
Thus, the correct answer is Option C.
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