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Answer :
Sure! Let's determine which expression is a prime polynomial.
A prime polynomial can't be factored over the integers (or the particular set of numbers we’re considering) into polynomials of lower degree, other than by multiplying by 1.
Let's analyze each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This expression has terms that share common factors. For example, [tex]\(10x^4\)[/tex], [tex]\(-5x^3\)[/tex], and [tex]\(70x^2\)[/tex] can be simplified by factoring common terms like [tex]\(5x\)[/tex].
- Since it can be factored, it is not a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This expression is a difference of cubes, as it fits the form [tex]\(a^3 - b^3\)[/tex] where [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex].
- Using the formula for the difference of cubes, [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex], it can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Therefore, it is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This is a quadratic expression in disguise. If we let [tex]\(u = x^2\)[/tex], the expression becomes [tex]\(u^2 + 20u - 100\)[/tex].
- Such a quadratic can be factored further, which means it isn’t prime.
D. [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by taking out the greatest common factor, 3, resulting in [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
Given these analyses, none of the given expressions are prime polynomials. Each one can be factored further.
A prime polynomial can't be factored over the integers (or the particular set of numbers we’re considering) into polynomials of lower degree, other than by multiplying by 1.
Let's analyze each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This expression has terms that share common factors. For example, [tex]\(10x^4\)[/tex], [tex]\(-5x^3\)[/tex], and [tex]\(70x^2\)[/tex] can be simplified by factoring common terms like [tex]\(5x\)[/tex].
- Since it can be factored, it is not a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This expression is a difference of cubes, as it fits the form [tex]\(a^3 - b^3\)[/tex] where [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex].
- Using the formula for the difference of cubes, [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex], it can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Therefore, it is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This is a quadratic expression in disguise. If we let [tex]\(u = x^2\)[/tex], the expression becomes [tex]\(u^2 + 20u - 100\)[/tex].
- Such a quadratic can be factored further, which means it isn’t prime.
D. [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by taking out the greatest common factor, 3, resulting in [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
Given these analyses, none of the given expressions are prime polynomials. Each one can be factored further.
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