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Answer :
To determine which expression is a prime polynomial, let's go through the options one by one. A prime polynomial is one that cannot be factored into the product of two or more non-constant polynomials.
### Expression A: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
1. Look for a common factor among all the terms. The common factor is [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 factored, this is not a prime polynomial.
### Expression B: [tex]\(x^3 - 27y^6\)[/tex]
1. Recognize the structure as a difference of cubes, since [tex]\(27y^6 = (3y^2)^3\)[/tex].
2. The difference of cubes formula: [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
3. Applying the formula: [tex]\(x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
4. Since it can be factored, this is not a prime polynomial.
### Expression C: [tex]\(3x^2 + 18y\)[/tex]
1. Look for a common factor, which is [tex]\(3\)[/tex].
2. Factoring out [tex]\(3\)[/tex], we get [tex]\(3(x^2 + 6y)\)[/tex].
3. Since it can be factored, this is not a prime polynomial.
### Expression D: [tex]\(x^4 + 20x^2 - 100\)[/tex]
1. This expression cannot be easily grouped or factored with basic methods.
2. After checking if any combinations or substitutions help in factoring, we'll find that it remains irreducible over the integers.
3. Since it cannot be factored into simpler polynomials with integer coefficients, it is considered a prime polynomial.
Based on this analysis, the expression that is a prime polynomial is Expression D: [tex]\(x^4 + 20x^2 - 100\)[/tex].
### Expression A: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
1. Look for a common factor among all the terms. The common factor is [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 factored, this is not a prime polynomial.
### Expression B: [tex]\(x^3 - 27y^6\)[/tex]
1. Recognize the structure as a difference of cubes, since [tex]\(27y^6 = (3y^2)^3\)[/tex].
2. The difference of cubes formula: [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
3. Applying the formula: [tex]\(x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
4. Since it can be factored, this is not a prime polynomial.
### Expression C: [tex]\(3x^2 + 18y\)[/tex]
1. Look for a common factor, which is [tex]\(3\)[/tex].
2. Factoring out [tex]\(3\)[/tex], we get [tex]\(3(x^2 + 6y)\)[/tex].
3. Since it can be factored, this is not a prime polynomial.
### Expression D: [tex]\(x^4 + 20x^2 - 100\)[/tex]
1. This expression cannot be easily grouped or factored with basic methods.
2. After checking if any combinations or substitutions help in factoring, we'll find that it remains irreducible over the integers.
3. Since it cannot be factored into simpler polynomials with integer coefficients, it is considered a prime polynomial.
Based on this analysis, the expression that is a prime polynomial is Expression D: [tex]\(x^4 + 20x^2 - 100\)[/tex].
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