Polynomials N zeros?
Hi,,
I wanted to ask why polynomials can have a maximum of N zeros.
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A polynomial over a body disintegrates into linear factors in its algebraic conclusion. If the basic ring is zero divider-free, it follows that there are no more than # linear factors= Deg polynomial zeros. In general, a polynomial can have more zerotels, for example, polynomials via Mat(n) or C(R), the quadratic dies via C/continuous function via R.
If a zero x_0 is present, the linear factor (x−x_0) can be split off with a polynomial division, whereby the degree of the polynomial is reduced by one. This can only be done as often as the degree of the original polynomial was.
Suppose a polynomial has n reelle zeros a1 . ., wherein individual zeros ak can be identical (multiple zero). The polynomial can then be represented as follows:
f(x) = b*(x – a1)*(x – a2)* … * (x – an)
When multiplying the n clamps, the highest potency is obtained to b*x^n, i.e. the polynomial is then also of degree n.
Only true about algebraic sealed bodies. For R, b is a product (x^2+|am^^2) and c .