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Answer :
Sure! Let's review what a prime polynomial is and evaluate the given expressions.
### What is a Prime Polynomial?
A prime polynomial is a polynomial that cannot be factored into polynomials of lower degree with coefficients in the same field (like real numbers, rational numbers, etc.). In simpler terms, it's similar to a prime number in arithmetic; it can only be "factored" by itself and 1.
### Evaluate Each Expression
To determine if an expression is a prime polynomial, let's analyze each option:
#### A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This polynomial is of degree 4. To check if it's prime, we would try to factor it.
However, upon testing for factorization, this expression is not prime, meaning it can be factored further.
#### B. [tex]\( x^3 - 27y^6 \)[/tex]
This polynomial is a difference of cubes:
[tex]\[ x^3 - (3y^2)^3 \][/tex]
We can use the formula for the difference of cubes:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
Let [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex]:
[tex]\[ x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + (3y^2)^2) = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
#### C. [tex]\( 3x^2 + 18y \)[/tex]
This polynomial is of degree 2. To check if it is prime, we look for common factors. We can factor out a 3:
[tex]\[ 3x^2 + 18y = 3(x^2 + 6y) \][/tex]
It is factored, hence not prime.
#### D. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
This polynomial is of degree 4. We look for any common factors first:
We can factor out an [tex]\( x \)[/tex]:
[tex]\[ 10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Thus, it can be factored further, so it is not a prime polynomial.
### Conclusion
After analyzing all the given polynomials, none of them qualify as prime polynomials because they can all be factored into lower-degree polynomials.
So, the answer is that none of the given polynomials are prime polynomials.
### What is a Prime Polynomial?
A prime polynomial is a polynomial that cannot be factored into polynomials of lower degree with coefficients in the same field (like real numbers, rational numbers, etc.). In simpler terms, it's similar to a prime number in arithmetic; it can only be "factored" by itself and 1.
### Evaluate Each Expression
To determine if an expression is a prime polynomial, let's analyze each option:
#### A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This polynomial is of degree 4. To check if it's prime, we would try to factor it.
However, upon testing for factorization, this expression is not prime, meaning it can be factored further.
#### B. [tex]\( x^3 - 27y^6 \)[/tex]
This polynomial is a difference of cubes:
[tex]\[ x^3 - (3y^2)^3 \][/tex]
We can use the formula for the difference of cubes:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
Let [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex]:
[tex]\[ x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + (3y^2)^2) = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
#### C. [tex]\( 3x^2 + 18y \)[/tex]
This polynomial is of degree 2. To check if it is prime, we look for common factors. We can factor out a 3:
[tex]\[ 3x^2 + 18y = 3(x^2 + 6y) \][/tex]
It is factored, hence not prime.
#### D. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
This polynomial is of degree 4. We look for any common factors first:
We can factor out an [tex]\( x \)[/tex]:
[tex]\[ 10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Thus, it can be factored further, so it is not a prime polynomial.
### Conclusion
After analyzing all the given polynomials, none of them qualify as prime polynomials because they can all be factored into lower-degree polynomials.
So, the answer is that none of the given polynomials are prime polynomials.
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