Answer :
We need to determine which of the following expressions is a prime (i.e., irreducible) polynomial, meaning it cannot be factored nontrivially into polynomials of lower degree with integer coefficients.
Let’s examine each option carefully:
1. Option A:
The expression is
[tex]$$3 x^2+18 y.$$[/tex]
Notice that there is a common factor of 3. Factoring it out gives:
[tex]$$3(x^2+6y).$$[/tex]
Since we have written it as a product of 3 (a unit factor up to multiplication by a constant) and another polynomial which is not a constant, this polynomial is not irreducible.
2. Option B:
The expression is
[tex]$$10 x^4-5 x^3+70 x^2+3 x.$$[/tex]
We can factor out the common factor [tex]$x$[/tex], which yields:
[tex]$$x(10x^3-5x^2+70x+3).$$[/tex]
Since the expression factors nontrivially (it is written as a product of [tex]$x$[/tex] and a cubic polynomial), it is not prime.
3. Option C:
The expression is
[tex]$$x^4+20x^2-100.$$[/tex]
On inspection, there does not appear to be any common factor or any factorization into lower-degree polynomials with integer coefficients. This expression remains in its given form and cannot be factored nontrivially over the integers. Therefore, it is irreducible (prime).
4. Option D:
The expression is
[tex]$$x^3-27y^6.$$[/tex]
Recognize that [tex]$27y^6$[/tex] can be written as [tex]$(3y^2)^3$[/tex]. Hence, the expression is a difference of cubes:
[tex]$$x^3-(3y^2)^3.$$[/tex]
The difference of cubes factors as:
[tex]$$a^3-b^3=(a-b)(a^2+ab+b^2).$$[/tex]
Applying this formula yields:
[tex]$$\Bigl(x-3y^2\Bigr)\Bigl(x^2+3xy^2+9y^4\Bigr),$$[/tex]
which is a nontrivial factorization. Thus, this polynomial is composite.
Based on this analysis, only Option C is prime.
Thus, the final answer is:
[tex]$$\boxed{x^4+20x^2-100.}$$[/tex]
Let’s examine each option carefully:
1. Option A:
The expression is
[tex]$$3 x^2+18 y.$$[/tex]
Notice that there is a common factor of 3. Factoring it out gives:
[tex]$$3(x^2+6y).$$[/tex]
Since we have written it as a product of 3 (a unit factor up to multiplication by a constant) and another polynomial which is not a constant, this polynomial is not irreducible.
2. Option B:
The expression is
[tex]$$10 x^4-5 x^3+70 x^2+3 x.$$[/tex]
We can factor out the common factor [tex]$x$[/tex], which yields:
[tex]$$x(10x^3-5x^2+70x+3).$$[/tex]
Since the expression factors nontrivially (it is written as a product of [tex]$x$[/tex] and a cubic polynomial), it is not prime.
3. Option C:
The expression is
[tex]$$x^4+20x^2-100.$$[/tex]
On inspection, there does not appear to be any common factor or any factorization into lower-degree polynomials with integer coefficients. This expression remains in its given form and cannot be factored nontrivially over the integers. Therefore, it is irreducible (prime).
4. Option D:
The expression is
[tex]$$x^3-27y^6.$$[/tex]
Recognize that [tex]$27y^6$[/tex] can be written as [tex]$(3y^2)^3$[/tex]. Hence, the expression is a difference of cubes:
[tex]$$x^3-(3y^2)^3.$$[/tex]
The difference of cubes factors as:
[tex]$$a^3-b^3=(a-b)(a^2+ab+b^2).$$[/tex]
Applying this formula yields:
[tex]$$\Bigl(x-3y^2\Bigr)\Bigl(x^2+3xy^2+9y^4\Bigr),$$[/tex]
which is a nontrivial factorization. Thus, this polynomial is composite.
Based on this analysis, only Option C is prime.
Thus, the final answer is:
[tex]$$\boxed{x^4+20x^2-100.}$$[/tex]
