Depending on its order and the number of possessed terms, polynomial factorization can be a lengthy and complicated process. The polynomial expression, (x2 − 2), is fortunately not one of those polynomials. The expression (x2 − 2) is a classic example of a difference of two squares. In factoring a difference of two squares, any expression in the form of (a2 − b2) is reduced to (a − b)(a + b). The key to this factoring process and ultimate solution for the expression (x2 − 2) lies in the square roots of its terms.
1. Calculating Square Roots
Calculate the square roots for 2 and x2. The square root of 2 is √2 and the square root of x2 is x.
2. Factoring the Polynomial
Write the equation
((x^2-2))
as the difference of two squares employing the terms’ square roots. You find that
((x^2-2) = (x-sqrt{2}) (x+sqrt{2}))
3. Solving the Equation
Set each expression in parentheses equal to 0, then solve. The first expression set to 0 yields
((x-sqrt{2})=0 text{ therefore } x= sqrt{2})
The second expression set to 0 yields
((x+ sqrt{2}) = 0 text{ therefore } x=- sqrt{2})
The solutions for x are √2 and −√2.
TL;DR (Too Long; Didn’t Read)
If needed, √2 can be converted into decimal form with a calculator, resulting in 1.41421356.