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adell [148]
3 years ago
10

What is the least possible degree of a polynomial that has roots -5,1 + 4i, and -4i?

Mathematics
1 answer:
Oksanka [162]3 years ago
6 0

Without any extra conditions, the answer could be 3, and the simplest polynomial with the given roots would be

(<em>x</em> + 5) (<em>x</em> - (1 + 4<em>i</em> )) (<em>x</em> + 4<em>i</em> )

= <em>x</em> ³ + 4<em>x</em> ² + (11 - 4<em>i</em> ) <em>x</em> + 80 - 2<em>i</em>

<em />

If the polynomial is supposed to have only <em>real</em> coefficients, then any complex roots must occur along with their complex conjugates:

(<em>x</em> + 5) (<em>x</em> - (1 + 4<em>i</em> )) (<em>x</em> - (1 - 4<em>i</em> )) (<em>x</em> + 4<em>i</em> ) (<em>x</em> - 4<em>i </em>)

= <em>x</em> ⁵ + 3<em>x</em> ⁴ + 23<em>x</em> ³ + 133<em>x</em> ² + 112<em>x</em> + 1360

and then the degree would be 5.

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