Hi Geri,

Let's try x² - 25 first.

x² - 25 = x² - 5² = (x - 5)(x + 5)

I know this because if you multiply (x - 5) by (x + 5) and simplify you get x² - 25. This means that for any number x, x² - 25 = (x - 5)(x + 5). For example if x = 12 then

12² - 5² = 144 - 25 = 119 = (12 - 5)(12 + 5) = 7 × 17.

Now what about x² + 25? If x = 2 for example then x² + 25 = 4 + 25 = 29 which is a prime number. Thus the only factors of 29 are 29 and 1 and hence the only factorization of 29 is 29 × 1. Similarly if x = 4 then x² + 25 = 41 which is again prime. Hence the only factorization of x² + 25 which is valid for every number x is

x² + 25 = (x² + 25) × 1

This is certainly not what you are looking for and thus we say x² + 25 doesn't factor.

Penny