Does a polynomial of degree n necessarily have n real solutions? Does it necessarily have n unique solutions, real or imaginary?

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Does a polynomial of degree n necessarily have n real solutions? Does it necessarily have n unique solutions, real or imaginary?

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Answer 1 · 2015-10-24T21:57:12

No. A polynomial equation in one variable of degree $n$ has exactly $n$ Complex roots, some of which may be Real, but some may be repeated roots.

Explanation:

For example, $0 = x^4+2x^2+1 = (x-i)^2(x+i)^2$ has roots $i$ , $i$ , $-i$ , $-i$ .

If the polynomial has Real coefficients, then any Complex roots occur as conjugate pairs. So if a polynomial of degree $n$ has Real coefficients, then it has $n - 2k$ Real roots and $2k$ non-Real Complex roots for some integer $k >= 0$ counting repeated roots.

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Does a polynomial of degree n necessarily have n real solutions? Does it necessarily have n unique solutions, real or imaginary?

1 Answer

Explanation:

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