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Answer :
We are given four polynomial expressions and need to determine which one is irreducible (i.e., it cannot be factored into a product of nonconstant polynomials with integer coefficients).
Step 1. Consider the expression
[tex]$$
x^3-27y^6.
$$[/tex]
Notice that it is a difference of cubes because
[tex]$$
27y^6 = (3y^2)^3.
$$[/tex]
Using the difference of cubes factorization, we write
[tex]$$
x^3- (3y^2)^3 = (x-3y^2)(x^2+3xy^2+9y^4).
$$[/tex]
Since it factors, this polynomial is reducible.
Step 2. Next, look at the expression
[tex]$$
3x^2+18y.
$$[/tex]
We can factor out the common factor 3:
[tex]$$
3x^2+18y=3(x^2+6y).
$$[/tex]
Because it can be written as a product where a non-unit (in the sense of a constant factor) is factored out, the polynomial is not considered prime in the context of finding an irreducible polynomial among univariate polynomials.
Step 3. Now consider the expression
[tex]$$
10x^4-5x^3+70x^2+3x.
$$[/tex]
In this expression, every term contains the variable [tex]$x$[/tex], so we can factor [tex]$x$[/tex] out:
[tex]$$
10x^4-5x^3+70x^2+3x = x(10x^3-5x^2+70x+3).
$$[/tex]
This shows that the polynomial is reducible.
Step 4. Finally, consider the expression
[tex]$$
x^4+20x^2-100.
$$[/tex]
This is a univariate polynomial in [tex]$x$[/tex]. One might try to factor it in the form
[tex]$$
(x^2+Ax+B)(x^2+Cx+D)
$$[/tex]
and compare coefficients. However, after performing such comparisons, it turns out that there is no factorization with all coefficients being integers. Hence, the polynomial does not factor into nonconstant polynomials with integer coefficients and is therefore irreducible.
Since only this polynomial is irreducible, the prime polynomial is
[tex]$$
\boxed{x^4+20x^2-100}.
$$[/tex]
Step 1. Consider the expression
[tex]$$
x^3-27y^6.
$$[/tex]
Notice that it is a difference of cubes because
[tex]$$
27y^6 = (3y^2)^3.
$$[/tex]
Using the difference of cubes factorization, we write
[tex]$$
x^3- (3y^2)^3 = (x-3y^2)(x^2+3xy^2+9y^4).
$$[/tex]
Since it factors, this polynomial is reducible.
Step 2. Next, look at the expression
[tex]$$
3x^2+18y.
$$[/tex]
We can factor out the common factor 3:
[tex]$$
3x^2+18y=3(x^2+6y).
$$[/tex]
Because it can be written as a product where a non-unit (in the sense of a constant factor) is factored out, the polynomial is not considered prime in the context of finding an irreducible polynomial among univariate polynomials.
Step 3. Now consider the expression
[tex]$$
10x^4-5x^3+70x^2+3x.
$$[/tex]
In this expression, every term contains the variable [tex]$x$[/tex], so we can factor [tex]$x$[/tex] out:
[tex]$$
10x^4-5x^3+70x^2+3x = x(10x^3-5x^2+70x+3).
$$[/tex]
This shows that the polynomial is reducible.
Step 4. Finally, consider the expression
[tex]$$
x^4+20x^2-100.
$$[/tex]
This is a univariate polynomial in [tex]$x$[/tex]. One might try to factor it in the form
[tex]$$
(x^2+Ax+B)(x^2+Cx+D)
$$[/tex]
and compare coefficients. However, after performing such comparisons, it turns out that there is no factorization with all coefficients being integers. Hence, the polynomial does not factor into nonconstant polynomials with integer coefficients and is therefore irreducible.
Since only this polynomial is irreducible, the prime polynomial is
[tex]$$
\boxed{x^4+20x^2-100}.
$$[/tex]
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