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Answer :
To determine which expression is a prime polynomial, we need to check if any of the provided polynomials can be factored further using integer coefficients. A prime polynomial cannot be factored into simpler polynomials with integer coefficients.
Let's evaluate each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- First, we can factor out a common term, [tex]\(x\)[/tex], from each part of the expression:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- This expression [tex]\(10x^3 - 5x^2 + 70x + 3\)[/tex] doesn't factor further using simple methods into polynomials with integer coefficients, but neither is it confirmed as prime without deeper analysis.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This expression is a difference of cubes:
[tex]\[
x^3 - (3y^2)^3
\][/tex]
- It can be factored using the formula for the difference of cubes:
[tex]\[
(x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- Since it can be factored, it is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This can be rewritten as a quadratic in terms of [tex]\(x^2\)[/tex]:
[tex]\[
(x^2)^2 + 20x^2 - 100
\][/tex]
- Let [tex]\(u = x^2\)[/tex], so it becomes [tex]\(u^2 + 20u - 100\)[/tex], which could potentially be factored further, thus it's not prime in its current form.
D. [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by taking out the greatest common factor, [tex]\(3\)[/tex]:
[tex]\[
3(x^2 + 6y)
\][/tex]
- Since it can be factored, it is not prime.
After reviewing all the options, we see that each one can be factored further or rewritten in a form suggesting it isn't necessarily prime. Generally, more context might be required to rule out potential simplifications further, but none of the given polynomials can be strictly confirmed as prime with the information at hand. Therefore, none of the given polynomials can be conclusively identified as a prime polynomial without additional context or constraints.
Let's evaluate each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- First, we can factor out a common term, [tex]\(x\)[/tex], from each part of the expression:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- This expression [tex]\(10x^3 - 5x^2 + 70x + 3\)[/tex] doesn't factor further using simple methods into polynomials with integer coefficients, but neither is it confirmed as prime without deeper analysis.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This expression is a difference of cubes:
[tex]\[
x^3 - (3y^2)^3
\][/tex]
- It can be factored using the formula for the difference of cubes:
[tex]\[
(x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- Since it can be factored, it is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This can be rewritten as a quadratic in terms of [tex]\(x^2\)[/tex]:
[tex]\[
(x^2)^2 + 20x^2 - 100
\][/tex]
- Let [tex]\(u = x^2\)[/tex], so it becomes [tex]\(u^2 + 20u - 100\)[/tex], which could potentially be factored further, thus it's not prime in its current form.
D. [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by taking out the greatest common factor, [tex]\(3\)[/tex]:
[tex]\[
3(x^2 + 6y)
\][/tex]
- Since it can be factored, it is not prime.
After reviewing all the options, we see that each one can be factored further or rewritten in a form suggesting it isn't necessarily prime. Generally, more context might be required to rule out potential simplifications further, but none of the given polynomials can be strictly confirmed as prime with the information at hand. Therefore, none of the given polynomials can be conclusively identified as a prime polynomial without additional context or constraints.
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