# MandysNotes

## The Roots of a Polynomial

09 March 2014 By Gideon
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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 $$\beta$$.