Answer :
A prime polynomial is one that cannot be factored into nontrivial factors (with integer or rational coefficients). We will analyze each option:
1. Consider the polynomial
[tex]$$P_A(x)=x^4+20x^2-100.$$[/tex]
One might view this as a quadratic in [tex]$x^2$[/tex], that is, if we set [tex]$u=x^2$[/tex], we have
[tex]$$u^2+20u-100.$$[/tex]
The discriminant of this quadratic is
[tex]$$20^2-4\cdot 1\cdot (-100)=400+400=800.$$[/tex]
Since [tex]$800$[/tex] is not a perfect square, the factors would involve irrational numbers. Therefore, the polynomial [tex]$P_A(x)$[/tex] cannot be factored into polynomials with integer or rational coefficients. This makes it a prime polynomial.
2. Now, examine
[tex]$$P_B(x,y)=x^3-27y^6.$$[/tex]
Recognize that [tex]$27y^6$[/tex] can be written as [tex]$(3y^2)^3$[/tex]. This is a difference of cubes:
[tex]$$x^3-(3y^2)^3,$$[/tex]
and it factors as
[tex]$$\left(x-3y^2\right)\left(x^2+3xy^2+9y^4\right).$$[/tex]
Since it factors nontrivially, it is not prime.
3. Next, look at
[tex]$$P_C(x)=10x^4-5x^3+70x^2+3x.$$[/tex]
We can factor out a common factor of [tex]$x$[/tex]:
[tex]$$x\left(10x^3-5x^2+70x+3\right).$$[/tex]
The presence of this factor clearly shows that [tex]$P_C(x)$[/tex] is composite.
4. Finally, consider
[tex]$$P_D(x,y)=3x^2+18y.$$[/tex]
A common factor of [tex]$3$[/tex] can be factored out:
[tex]$$3\left(x^2+6y\right).$$[/tex]
This factorization also shows that it is not prime.
Since the only expression that does not factor (with integer or rational coefficients) is
[tex]$$x^4+20x^2-100,$$[/tex]
the prime polynomial is given in Option A.
Thus, the correct answer is Option A.
1. Consider the polynomial
[tex]$$P_A(x)=x^4+20x^2-100.$$[/tex]
One might view this as a quadratic in [tex]$x^2$[/tex], that is, if we set [tex]$u=x^2$[/tex], we have
[tex]$$u^2+20u-100.$$[/tex]
The discriminant of this quadratic is
[tex]$$20^2-4\cdot 1\cdot (-100)=400+400=800.$$[/tex]
Since [tex]$800$[/tex] is not a perfect square, the factors would involve irrational numbers. Therefore, the polynomial [tex]$P_A(x)$[/tex] cannot be factored into polynomials with integer or rational coefficients. This makes it a prime polynomial.
2. Now, examine
[tex]$$P_B(x,y)=x^3-27y^6.$$[/tex]
Recognize that [tex]$27y^6$[/tex] can be written as [tex]$(3y^2)^3$[/tex]. This is a difference of cubes:
[tex]$$x^3-(3y^2)^3,$$[/tex]
and it factors as
[tex]$$\left(x-3y^2\right)\left(x^2+3xy^2+9y^4\right).$$[/tex]
Since it factors nontrivially, it is not prime.
3. Next, look at
[tex]$$P_C(x)=10x^4-5x^3+70x^2+3x.$$[/tex]
We can factor out a common factor of [tex]$x$[/tex]:
[tex]$$x\left(10x^3-5x^2+70x+3\right).$$[/tex]
The presence of this factor clearly shows that [tex]$P_C(x)$[/tex] is composite.
4. Finally, consider
[tex]$$P_D(x,y)=3x^2+18y.$$[/tex]
A common factor of [tex]$3$[/tex] can be factored out:
[tex]$$3\left(x^2+6y\right).$$[/tex]
This factorization also shows that it is not prime.
Since the only expression that does not factor (with integer or rational coefficients) is
[tex]$$x^4+20x^2-100,$$[/tex]
the prime polynomial is given in Option A.
Thus, the correct answer is Option A.
