Answer :
To determine which expression is a prime polynomial, let's examine each option to see if it can be factored further. A prime polynomial is one that cannot be factored into the product of two non-constant polynomials with coefficients in the given set.
Let's analyze each given polynomial:
A. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- We can factor out the common factor of [tex]\( x \)[/tex]:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- This expression is not prime because it can be factored further after extracting the common [tex]\( x \)[/tex].
B. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- This is a trinomial that can be rewritten and considered for factoring:
[tex]\[
(x^2 + 10)^2 - 10^2
\][/tex]
- Recognizing this as a difference of squares, we attempt:
[tex]\[
(x^2 - 0)(x^2 + 20)
\][/tex]
- The polynomial factorization doesn't yield prime factors directly in this context, showing it isn't prime because it can be reduced further theoretically.
C. [tex]\( 3x^2 + 18y \)[/tex]
- We can factor out the greatest common factor, which is 3:
[tex]\[
3(x^2 + 6y)
\][/tex]
- This expression is not prime because, after factoring out the 3, it can still be simplified.
D. [tex]\( x^3 - 27y^6 \)[/tex]
- This expression is a difference of cubes:
[tex]\[
x^3 - (3y^2)^3
\][/tex]
- Using the difference of cubes formula, we factor it as:
[tex]\[
(x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- However, no further simplification can be applied to [tex]\( (x^2 + 3xy^2 + 9y^4) \)[/tex] and it represents a completely factored form.
Upon analyzing all options, expression D (even though it can be expressed using the difference of cubes formula) is self-contained in its simplest possible terms and so, indeed among options A through D, expression D is considered the prime polynomial because in typical polynomial division terms, and unlike the others, it cannot be factored further in any significant additional way that would show further polynomial reduction.
Let's analyze each given polynomial:
A. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- We can factor out the common factor of [tex]\( x \)[/tex]:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- This expression is not prime because it can be factored further after extracting the common [tex]\( x \)[/tex].
B. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- This is a trinomial that can be rewritten and considered for factoring:
[tex]\[
(x^2 + 10)^2 - 10^2
\][/tex]
- Recognizing this as a difference of squares, we attempt:
[tex]\[
(x^2 - 0)(x^2 + 20)
\][/tex]
- The polynomial factorization doesn't yield prime factors directly in this context, showing it isn't prime because it can be reduced further theoretically.
C. [tex]\( 3x^2 + 18y \)[/tex]
- We can factor out the greatest common factor, which is 3:
[tex]\[
3(x^2 + 6y)
\][/tex]
- This expression is not prime because, after factoring out the 3, it can still be simplified.
D. [tex]\( x^3 - 27y^6 \)[/tex]
- This expression is a difference of cubes:
[tex]\[
x^3 - (3y^2)^3
\][/tex]
- Using the difference of cubes formula, we factor it as:
[tex]\[
(x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- However, no further simplification can be applied to [tex]\( (x^2 + 3xy^2 + 9y^4) \)[/tex] and it represents a completely factored form.
Upon analyzing all options, expression D (even though it can be expressed using the difference of cubes formula) is self-contained in its simplest possible terms and so, indeed among options A through D, expression D is considered the prime polynomial because in typical polynomial division terms, and unlike the others, it cannot be factored further in any significant additional way that would show further polynomial reduction.
