To determine whether the given polynomial [tex]\(x^2 + 25y^2\)[/tex] can be factored, let's first examine the general form and characteristics of the polynomial.
The polynomial in question is [tex]\(x^2 + 25y^2\)[/tex].
1. Form of the Polynomial:
The given polynomial [tex]\(x^2 + 25y^2\)[/tex] is a sum of two squares. Recall that:
- In algebra, a sum of squares [tex]\(a^2 + b^2\)[/tex] does not generally have a factorization over the real numbers (or integers) unless it can be further simplified or has any special properties.
2. Checking Possible Factors:
We need to check if the polynomial can be written as a product of two binomials. Consider the following options:
- [tex]\((x + 5y)(x + 5y)\)[/tex]
- [tex]\((x + 5y)(x - 5y)\)[/tex]
- [tex]\((x - 5y)(x - 5y)\)[/tex]
3. Evaluate Each Option:
- [tex]\((x + 5y)(x + 5y)\)[/tex]:
[tex]\[
(x + 5y)^2 = x^2 + 2 \cdot x \cdot 5y + (5y)^2 = x^2 + 10xy + 25y^2
\][/tex]
This results in [tex]\(x^2 + 10xy + 25y^2\)[/tex], not our original polynomial.
- [tex]\((x + 5y)(x - 5y)\)[/tex]:
[tex]\[
(x + 5y)(x - 5y) = x^2 - (5y)^2 = x^2 - 25y^2
\][/tex]
This results in [tex]\(x^2 - 25y^2\)[/tex], which again differs from our original polynomial.
- [tex]\((x - 5y)(x - 5y)\)[/tex]:
[tex]\[
(x - 5y)^2 = x^2 - 2 \cdot x \cdot 5y + (5y)^2 = x^2 - 10xy + 25y^2
\][/tex]
This results in [tex]\(x^2 - 10xy + 25y^2\)[/tex], not matching our original polynomial.
4. Conclusion:
Since none of the binomial pair options yield [tex]\(x^2 + 25y^2\)[/tex] when expanded, and because a sum of squares like this generally cannot be factored over the integers, we conclude that [tex]\(x^2 + 25y^2\)[/tex] cannot be factored. Therefore, the polynomial is prime.
Thus, [tex]\(x^2 + 25y^2\)[/tex] is Prime.
