# 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

QUESTION POSTED 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]

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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