How do you factor the expression 25x^2 + 1?

To factor the expression 25x² + 1, we first notice that this is a sum of squares. The sum of squares, like a² + b², does not factor nicely over the real numbers. However, we can express it in another form using complex numbers.

Let’s rewrite the expression:

25x² + 1 = (5x)² + 1²

Now, according to the sum of squares formula, a² + b² = (a + bi)(a – bi). Here, we can identify:

• a = 5x
• b = 1

Using this, we can factor:

25x² + 1 = (5x + i)(5x – i)

In this factorization, i is the imaginary unit, where i = √-1. Thus, while the expression 25x² + 1 doesn’t have real roots, we can represent it as a product of two complex factors.