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Answer :
To determine which expression is a prime polynomial, let's analyze each option.
A prime polynomial is one that cannot be factored into simpler polynomials with integer coefficients.
Option A: [tex]\(3x^2 + 18y\)[/tex]
1. Look for a common factor in the terms.
2. We can factor out a 3: [tex]\(3(x^2 + 6y)\)[/tex].
3. Since it can be factored, it's not a prime polynomial.
Option B: [tex]\(x^3 - 27y^6\)[/tex]
1. Recognize this as a difference of cubes.
2. The difference of cubes formula is [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
3. Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex], so it factors to: [tex]\((x - 3y^2)(x^2 + 3xy^2 + (3y^2)^2)\)[/tex].
4. Because it can be factored, it is not a prime polynomial.
Option C: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
1. Look for a common factor in every term.
2. The greatest common factor is [tex]\(x\)[/tex]: [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
3. Since it can be factored, it's not a prime polynomial.
Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex]
1. Consider using a substitution: let [tex]\(y = x^2\)[/tex], which transforms the expression to [tex]\(y^2 + 20y - 100\)[/tex].
2. To see if [tex]\(y^2 + 20y - 100\)[/tex] can be factored, check if there are factors of -100 that add up to 20. There are none that meet this condition with integer values.
3. Since the polynomial cannot be factored over the integers, [tex]\(x^4 + 20x^2 - 100\)[/tex] is a prime polynomial.
Therefore, the prime polynomial is Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex].
A prime polynomial is one that cannot be factored into simpler polynomials with integer coefficients.
Option A: [tex]\(3x^2 + 18y\)[/tex]
1. Look for a common factor in the terms.
2. We can factor out a 3: [tex]\(3(x^2 + 6y)\)[/tex].
3. Since it can be factored, it's not a prime polynomial.
Option B: [tex]\(x^3 - 27y^6\)[/tex]
1. Recognize this as a difference of cubes.
2. The difference of cubes formula is [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
3. Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex], so it factors to: [tex]\((x - 3y^2)(x^2 + 3xy^2 + (3y^2)^2)\)[/tex].
4. Because it can be factored, it is not a prime polynomial.
Option C: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
1. Look for a common factor in every term.
2. The greatest common factor is [tex]\(x\)[/tex]: [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
3. Since it can be factored, it's not a prime polynomial.
Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex]
1. Consider using a substitution: let [tex]\(y = x^2\)[/tex], which transforms the expression to [tex]\(y^2 + 20y - 100\)[/tex].
2. To see if [tex]\(y^2 + 20y - 100\)[/tex] can be factored, check if there are factors of -100 that add up to 20. There are none that meet this condition with integer values.
3. Since the polynomial cannot be factored over the integers, [tex]\(x^4 + 20x^2 - 100\)[/tex] is a prime polynomial.
Therefore, the prime polynomial is Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex].
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