Factoring the expression x² + 81 involves recognizing that this is a sum of squares, which doesn’t factor over the real numbers in a straightforward way. However, we can express this expression in terms of complex numbers.
To factor x² + 81, we note that 81 can be written as (9)². Thus, we can rewrite the expression as:
x² + (9)²
The sum of squares can be factored using the identity:
a² + b² = (a + bi)(a – bi)
In this case, a is x and b is 9. Therefore, we can apply the identity:
x² + 81 = (x + 9i)(x – 9i)
So, the complete factorization of x² + 81 in terms of complex numbers is:
(x + 9i)(x – 9i)
In summary, while x² + 81 does not factor nicely over the reals, it can be expressed as a product of two complex factors using the sum of squares identity.