Answer :
We are given four expressions and asked to determine which one is a prime polynomial (one that cannot be factored into nontrivial polynomials with integer coefficients).
Let’s analyze each option:
1. Option A:
The expression is
[tex]$$10x^4 - 5x^3 + 70x^2 + 3x.$$[/tex]
Notice that every term contains an [tex]$x$[/tex], so we can factor it out:
[tex]$$x(10x^3 - 5x^2 + 70x + 3).$$[/tex]
Since there is a common factor ([tex]$x$[/tex]), the polynomial is not prime.
2. Option B:
The expression is
[tex]$$x^4 + 20x^2 - 100.$$[/tex]
To check its factorability, one approach is to use a substitution. Let [tex]$u = x^2$[/tex]. Under this substitution, the polynomial becomes:
[tex]$$u^2 + 20u - 100.$$[/tex]
We examine the quadratic in [tex]$u$[/tex]. Its discriminant is
[tex]$$\Delta = 20^2 - 4(1)(-100) = 400 + 400 = 800.$$[/tex]
The discriminant, [tex]$800$[/tex], is not a perfect square, which indicates that the quadratic does not factor over the integers. Attempts to factor it into the product of two quadratic expressions with integer coefficients will not be successful. Therefore, this expression cannot be factored into simpler polynomials with integer coefficients and is prime.
3. Option C:
The expression is
[tex]$$x^3 - 27y^6.$$[/tex]
Recognize this as a difference of cubes, since [tex]$27y^6 = (3y^2)^3$[/tex]. Using the difference of cubes formula
[tex]$$a^3 - b^3 = (a - b)(a^2 + ab + b^2),$$[/tex]
with [tex]$a = x$[/tex] and [tex]$b = 3y^2$[/tex], we factor the expression as:
[tex]$$\left(x - 3y^2\right)\left(x^2 + 3xy^2 + 9y^4\right).$$[/tex]
Hence, this polynomial is not prime.
4. Option D:
The expression is
[tex]$$3x^2 + 18y.$$[/tex]
We notice that both terms share a common factor of [tex]$3$[/tex], so we factor it out:
[tex]$$3(x^2 + 6y).$$[/tex]
This shows that the polynomial has a nontrivial factorization, and thus it is not prime.
Since Option B does not have any factorization into lower-degree polynomials with integer coefficients, it is the prime polynomial.
The correct answer is Option B.
Let’s analyze each option:
1. Option A:
The expression is
[tex]$$10x^4 - 5x^3 + 70x^2 + 3x.$$[/tex]
Notice that every term contains an [tex]$x$[/tex], so we can factor it out:
[tex]$$x(10x^3 - 5x^2 + 70x + 3).$$[/tex]
Since there is a common factor ([tex]$x$[/tex]), the polynomial is not prime.
2. Option B:
The expression is
[tex]$$x^4 + 20x^2 - 100.$$[/tex]
To check its factorability, one approach is to use a substitution. Let [tex]$u = x^2$[/tex]. Under this substitution, the polynomial becomes:
[tex]$$u^2 + 20u - 100.$$[/tex]
We examine the quadratic in [tex]$u$[/tex]. Its discriminant is
[tex]$$\Delta = 20^2 - 4(1)(-100) = 400 + 400 = 800.$$[/tex]
The discriminant, [tex]$800$[/tex], is not a perfect square, which indicates that the quadratic does not factor over the integers. Attempts to factor it into the product of two quadratic expressions with integer coefficients will not be successful. Therefore, this expression cannot be factored into simpler polynomials with integer coefficients and is prime.
3. Option C:
The expression is
[tex]$$x^3 - 27y^6.$$[/tex]
Recognize this as a difference of cubes, since [tex]$27y^6 = (3y^2)^3$[/tex]. Using the difference of cubes formula
[tex]$$a^3 - b^3 = (a - b)(a^2 + ab + b^2),$$[/tex]
with [tex]$a = x$[/tex] and [tex]$b = 3y^2$[/tex], we factor the expression as:
[tex]$$\left(x - 3y^2\right)\left(x^2 + 3xy^2 + 9y^4\right).$$[/tex]
Hence, this polynomial is not prime.
4. Option D:
The expression is
[tex]$$3x^2 + 18y.$$[/tex]
We notice that both terms share a common factor of [tex]$3$[/tex], so we factor it out:
[tex]$$3(x^2 + 6y).$$[/tex]
This shows that the polynomial has a nontrivial factorization, and thus it is not prime.
Since Option B does not have any factorization into lower-degree polynomials with integer coefficients, it is the prime polynomial.
The correct answer is Option B.
