The roots of the quadratic polynomial defined by the equation

\[(x - \alpha)(x - \beta) = x^{2} - (\alpha + \beta)x + \alpha\beta.\]

are \(\alpha\) and \(\beta\).

**Proof:**

By direct substitution into equation the above equation we see that:

- \(P(\alpha) = (\alpha - \alpha)(\alpha - \beta) = 0;\)
- \(P(\beta) = (\beta - \alpha)(\beta - \beta) = 0.\)

Now suppose there were some other value \(x=\gamma\) such that \(P(\gamma) = 0,\) then:

\[ (\gamma - \alpha) = c_1 \neq 0, \]

and

\[(\gamma - \beta) = c_2 \neq 0,\]

but if \(P(\gamma) =0\) then,

\[ P(\gamma) = (\gamma - \alpha)(\gamma - \beta) = c_1 c_2 = 0.\]

But two nonzero real numbers cannot multiply to zero because there are no zero divisors in the field of real numbers.

Therefore \(\gamma\) must equal either \(\alpha\) or \(\beta\).