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Answer :
To determine which expression is a prime polynomial, we need to analyze whether each polynomial can be factored into simpler polynomials.
1. A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Check if there is a common factor. We can see that 5x is a common factor in the first three terms. However, the last term, 3x, does not fit neatly into a simple factorization with the others. Thus, it is not easily factored into simpler polynomials beyond removing the common factor, making it prime.
2. B. [tex]\(3x^2 + 18y\)[/tex]
- Notice that both terms have a common factor of 3.
- This can be factored as [tex]\(3(x^2 + 6y)\)[/tex]. Therefore, this is not a prime polynomial.
3. C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This is a quadratic in disguise (you can let [tex]\(z = x^2\)[/tex] and rewrite it as [tex]\(z^2 + 20z - 100\)[/tex]).
- It can potentially be factored using various methods for solving quadratic equations, such as factoring by grouping or using the quadratic formula. Therefore, this is not a prime polynomial.
4. D. [tex]\(x^3 - 27y^6\)[/tex]
- This expression is a difference of cubes.
- It can be factored using the formula [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = y^2\)[/tex].
- Therefore, this can be factored, making it non-prime.
After this analysis, the correct expression that is a prime polynomial is:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
1. A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Check if there is a common factor. We can see that 5x is a common factor in the first three terms. However, the last term, 3x, does not fit neatly into a simple factorization with the others. Thus, it is not easily factored into simpler polynomials beyond removing the common factor, making it prime.
2. B. [tex]\(3x^2 + 18y\)[/tex]
- Notice that both terms have a common factor of 3.
- This can be factored as [tex]\(3(x^2 + 6y)\)[/tex]. Therefore, this is not a prime polynomial.
3. C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This is a quadratic in disguise (you can let [tex]\(z = x^2\)[/tex] and rewrite it as [tex]\(z^2 + 20z - 100\)[/tex]).
- It can potentially be factored using various methods for solving quadratic equations, such as factoring by grouping or using the quadratic formula. Therefore, this is not a prime polynomial.
4. D. [tex]\(x^3 - 27y^6\)[/tex]
- This expression is a difference of cubes.
- It can be factored using the formula [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = y^2\)[/tex].
- Therefore, this can be factored, making it non-prime.
After this analysis, the correct expression that is a prime polynomial is:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
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