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Answer :
To determine which expression is a prime polynomial, we check if each polynomial can be factored into simpler polynomials with integer coefficients. A prime polynomial cannot be broken down into such factors, similar to a prime number.
Let's go through each option:
A. [tex]\(3x^2 + 18y\)[/tex]
1. Notice that this expression has a common factor of 3.
2. Factoring out 3, we get [tex]\(3(x^2 + 6y)\)[/tex].
3. Since it can be broken down into a multiple of 3 and a simpler polynomial, [tex]\(3x^2 + 18y\)[/tex] is not prime.
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]
1. This polynomial appears to be in the form of a quadratic in terms of [tex]\(x^2\)[/tex].
2. We try to factor it further:
- Let [tex]\(z = x^2\)[/tex].
- The polynomial becomes [tex]\(z^2 + 20z - 100\)[/tex].
3. We can use the quadratic formula here, but without specific methods, this polynomial does not have simple integer factors.
4. Since it can't be factored into polynomials with integer coefficients, this polynomial is considered prime.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
1. Notice that this expression has a common factor of [tex]\(x\)[/tex].
2. Factoring out [tex]\(x\)[/tex], we get [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
3. Since it can be broken down with a factor of [tex]\(x\)[/tex], it is not prime.
D. [tex]\(x^3 - 27y^6\)[/tex]
1. This is a difference of cubes: [tex]\(x^3 - (3y^2)^3\)[/tex].
2. We can apply the difference of cubes formula: [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
3. For this polynomial: [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex].
4. The expression can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
5. Since we can factor this polynomial, it is not prime.
Based on these steps, the expression that is a prime polynomial is:
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Let's go through each option:
A. [tex]\(3x^2 + 18y\)[/tex]
1. Notice that this expression has a common factor of 3.
2. Factoring out 3, we get [tex]\(3(x^2 + 6y)\)[/tex].
3. Since it can be broken down into a multiple of 3 and a simpler polynomial, [tex]\(3x^2 + 18y\)[/tex] is not prime.
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]
1. This polynomial appears to be in the form of a quadratic in terms of [tex]\(x^2\)[/tex].
2. We try to factor it further:
- Let [tex]\(z = x^2\)[/tex].
- The polynomial becomes [tex]\(z^2 + 20z - 100\)[/tex].
3. We can use the quadratic formula here, but without specific methods, this polynomial does not have simple integer factors.
4. Since it can't be factored into polynomials with integer coefficients, this polynomial is considered prime.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
1. Notice that this expression has a common factor of [tex]\(x\)[/tex].
2. Factoring out [tex]\(x\)[/tex], we get [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
3. Since it can be broken down with a factor of [tex]\(x\)[/tex], it is not prime.
D. [tex]\(x^3 - 27y^6\)[/tex]
1. This is a difference of cubes: [tex]\(x^3 - (3y^2)^3\)[/tex].
2. We can apply the difference of cubes formula: [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
3. For this polynomial: [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex].
4. The expression can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
5. Since we can factor this polynomial, it is not prime.
Based on these steps, the expression that is a prime polynomial is:
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]
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