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Answer :
To determine which expression is a prime polynomial, let's first define what a prime polynomial is. A prime polynomial is one that cannot be factored into a product of polynomials with integer coefficients that have lower degrees.
Let's examine each option:
A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This polynomial can potentially be factored using polynomial factorization techniques. However, upon attempting to factor it, it cannot be broken down further into polynomials of lower degrees with integer coefficients. Therefore, option A is likely to be a prime polynomial.
B. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
This polynomial can be factored at least by taking [tex]\( x \)[/tex] common, resulting in [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex]. Therefore, it is not a prime polynomial because it can be factored into parts with lower degrees.
C. [tex]\( x^3 - 27y^6 \)[/tex]
This expression appears to be a difference of cubes: [tex]\( x^3 - (3y^2)^3 \)[/tex]. It can be factored using the difference of cubes formula:
[tex]\[ x^3 - a^3 = (x - a)(x^2 + ax + a^2) \][/tex]
Here, it factors to:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored further, it's not a prime polynomial.
D. [tex]\( 3x^2 + 18y \)[/tex]
This polynomial can be factored by taking out the greatest common factor, which is 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored, it is not a prime polynomial.
Considering the options, the expression that cannot be factored further using polynomial factorization with integer coefficients is:
A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This means that expression A is a prime polynomial.
Let's examine each option:
A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This polynomial can potentially be factored using polynomial factorization techniques. However, upon attempting to factor it, it cannot be broken down further into polynomials of lower degrees with integer coefficients. Therefore, option A is likely to be a prime polynomial.
B. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
This polynomial can be factored at least by taking [tex]\( x \)[/tex] common, resulting in [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex]. Therefore, it is not a prime polynomial because it can be factored into parts with lower degrees.
C. [tex]\( x^3 - 27y^6 \)[/tex]
This expression appears to be a difference of cubes: [tex]\( x^3 - (3y^2)^3 \)[/tex]. It can be factored using the difference of cubes formula:
[tex]\[ x^3 - a^3 = (x - a)(x^2 + ax + a^2) \][/tex]
Here, it factors to:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored further, it's not a prime polynomial.
D. [tex]\( 3x^2 + 18y \)[/tex]
This polynomial can be factored by taking out the greatest common factor, which is 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored, it is not a prime polynomial.
Considering the options, the expression that cannot be factored further using polynomial factorization with integer coefficients is:
A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This means that expression A is a prime polynomial.
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