Answer :
We need to determine which of the given expressions does not factor nontrivially into lower-degree polynomial factors over the integers. In other words, we want to find the prime polynomial.
Let’s examine each option:
1. Option A:
[tex]$$
10x^4 - 5x^3 + 70x^2 + 3x
$$[/tex]
Notice that every term contains a factor of [tex]$x$[/tex]. Factoring out [tex]$x$[/tex], we get:
[tex]$$
x(10x^3 - 5x^2 + 70x + 3)
$$[/tex]
Since it factors as a product including [tex]$x$[/tex], this expression is composite.
2. Option B:
[tex]$$
x^4 + 20x^2 - 100
$$[/tex]
Although this polynomial might look factorable at first glance, one way to check is to perform a substitution such as [tex]$y=x^2$[/tex]. This gives:
[tex]$$
y^2 + 20y - 100
$$[/tex]
The discriminant of the quadratic in [tex]$y$[/tex] is:
[tex]$$
\Delta = 20^2 - 4(1)(-100) = 400 + 400 = 800
$$[/tex]
Since [tex]$\Delta = 800$[/tex] is not a perfect square, the quadratic does not factor over the integers. Thus the original polynomial does not factor into lower degree polynomials with integer coefficients and is prime.
3. Option C:
[tex]$$
3x^2 + 18y
$$[/tex]
Here, a common factor of [tex]$3$[/tex] can be factored out:
[tex]$$
3(x^2 + 6y)
$$[/tex]
This shows the expression is composite.
4. Option D:
[tex]$$
x^3 - 27y^6
$$[/tex]
Recognize this as a difference of cubes because [tex]$27y^6 = (3y^2)^3$[/tex]. Thus, it factors as:
[tex]$$
x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
$$[/tex]
This factorization confirms that the expression is composite.
Since only Option B remains unfactored (prime), the correct answer is Option B.
Thus, the prime polynomial is:
[tex]$$
x^4 + 20x^2 - 100
$$[/tex]
If answers are numbered by the order in which the options are given, the answer is [tex]$\boxed{2}$[/tex].
Let’s examine each option:
1. Option A:
[tex]$$
10x^4 - 5x^3 + 70x^2 + 3x
$$[/tex]
Notice that every term contains a factor of [tex]$x$[/tex]. Factoring out [tex]$x$[/tex], we get:
[tex]$$
x(10x^3 - 5x^2 + 70x + 3)
$$[/tex]
Since it factors as a product including [tex]$x$[/tex], this expression is composite.
2. Option B:
[tex]$$
x^4 + 20x^2 - 100
$$[/tex]
Although this polynomial might look factorable at first glance, one way to check is to perform a substitution such as [tex]$y=x^2$[/tex]. This gives:
[tex]$$
y^2 + 20y - 100
$$[/tex]
The discriminant of the quadratic in [tex]$y$[/tex] is:
[tex]$$
\Delta = 20^2 - 4(1)(-100) = 400 + 400 = 800
$$[/tex]
Since [tex]$\Delta = 800$[/tex] is not a perfect square, the quadratic does not factor over the integers. Thus the original polynomial does not factor into lower degree polynomials with integer coefficients and is prime.
3. Option C:
[tex]$$
3x^2 + 18y
$$[/tex]
Here, a common factor of [tex]$3$[/tex] can be factored out:
[tex]$$
3(x^2 + 6y)
$$[/tex]
This shows the expression is composite.
4. Option D:
[tex]$$
x^3 - 27y^6
$$[/tex]
Recognize this as a difference of cubes because [tex]$27y^6 = (3y^2)^3$[/tex]. Thus, it factors as:
[tex]$$
x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
$$[/tex]
This factorization confirms that the expression is composite.
Since only Option B remains unfactored (prime), the correct answer is Option B.
Thus, the prime polynomial is:
[tex]$$
x^4 + 20x^2 - 100
$$[/tex]
If answers are numbered by the order in which the options are given, the answer is [tex]$\boxed{2}$[/tex].
