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The Roots of a Polynomial

09 March 2014 By Gideon In Polynomials
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Definition:

The roots of a polynomial \(P\) are the solutions of the equation:

\[ P(x) = 0. \]


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 \(\beta\).

Read 1689 times Last modified on Sunday, 09 March 2014 17:07

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