Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree​ 4; ​ zeros −4−5i; −2 multiplicity 2

QUESTION POSTED AT 01/06/2020 - 02:59 PM

Answered by admin AT 01/06/2020 - 02:59 PM

There are many polynomials that fit the bill,
f(x)=a(x-r1)(x-r2)(x-r3)(x-r4)  where a is any real number not equal to zero.
A simple one is when a=1.
where r1,r2,r3,r4 are the roots of the 4th degree polynomial.
Also note that for a polynomial with *real* coefficients, complex roots *always* come in conjugages, i.e. in the form a±bi  [±=+/-]

So a polynomial would be:
f(x)=(x-(-4-5i))(x-(-4+5i))(x--2)(x--2)
or, simplifying
f(x)=(x+4+5i)(x+4-5i)(x+2)^2
=x^4+12x^3+77x^2+196x+164   [if you decide to expand]
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