To determine if the expression [tex]\(x^2 + 25y^2\)[/tex] can be factored, let us first understand the nature of this expression.
The expression [tex]\(x^2 + 25y^2\)[/tex] consists of two terms: [tex]\(x^2\)[/tex] (which is a perfect square) and [tex]\(25y^2\)[/tex] (which is also a perfect square since it can be written as [tex]\((5y)^2\)[/tex]). However, the expression is a sum of squares, not a difference of squares.
In algebra, a sum of squares does not factor into real polynomial factors. In other words, the expression [tex]\(x^2 + (5y)^2\)[/tex] cannot be decomposed into a product of binomials with real coefficients.
The provided options for factoring are:
1. [tex]\((x + 5y)(x + 5y)\)[/tex]
2. [tex]\((x + 5y)(x - 5y)\)[/tex]
3. [tex]\((x - 5y)(x - 5y)\)[/tex]
4. Prime
These options suggest possible factorizations, but let us verify each:
1. [tex]\((x + 5y)(x + 5y)\)[/tex] - This would expand to [tex]\(x^2 + 10xy + 25y^2\)[/tex], not [tex]\(x^2 + 25y^2\)[/tex].
2. [tex]\((x + 5y)(x - 5y)\)[/tex] - This would expand to [tex]\(x^2 - 25y^2\)[/tex], which is a difference of squares, not a sum of squares.
3. [tex]\((x - 5y)(x - 5y)\)[/tex] - This would expand to [tex]\(x^2 - 10xy + 25y^2\)[/tex], not [tex]\(x^2 + 25y^2\)[/tex].
None of these options successfully replicate the original expression.
Since [tex]\(x^2 + 25y^2\)[/tex] cannot be factored into real polynomial factors, it is considered a prime polynomial.
Therefore, the correct answer is:
- Prime
