Factoring the Expression 16x² + 25
The expression 16x² + 25 is a sum of squares, which does not factor into real numbers using traditional methods. However, we can attempt to express it using complex numbers.
Recognizing the Form
The expression can be recognized as:
- A² + B²
where:
- A = 4x
- B = 5
Using the Sum of Squares Formula
In general, the sum of two squares can be expressed in complex factor form. The algebraic identity states:
- A² + B² = (A + Bi)(A – Bi)
Applying this to our expression:
- Let A = 4x and B = 5.
- Then, substituting into the formula, we have:
16x² + 25 = (4x + 5i)(4x – 5i)
Conclusion
Thus, the complete factorization of the expression 16x² + 25 over the complex numbers is:
(4x + 5i)(4x – 5i)
It’s important to note that this factorization involves imaginary numbers, indicating that the expression does not have real roots, as a sum of squares typically does not factor nicely in the realm of real numbers alone.