Question

write the simplest polynomial function with the zeros 2-i,√5, and -2.

Answer 1

hint:
(x-(2-isqrt5))*(x-(-2))=?

Answer 2

In general, the simplest polynomial function with zeros,
\([\rm part~1]\) \(\color{#000000 }{ \displaystyle a }\)
\([\rm part~2]\) \(\color{#000000 }{ \displaystyle a,~b }\)
\([\rm part~3]\) \(\color{#000000 }{ \displaystyle a,~b,~{\tt \small and}~c }\)
\(\rm And~so~forth{\bf~.....}\)

is going to be:
\([\rm part~1]\) \(\color{#000000 }{ \displaystyle f(x)=x-a }\)
\([\rm part~2]\) \(\color{#000000 }{ \displaystyle f(x)=(x-a)(x-b) }\)
\([\rm part~3]\) \(\color{#000000 }{ \displaystyle f(x)=(x-a)(x-b)(x-c) }\)

Reasoning/because
\([\rm part~1]\) \(\color{#000000 }{ \displaystyle f(x)=0 }\) when evaluated at \(\color{#000000 }{ \displaystyle x=a }\)
\([\rm part~2]\) \(\color{#000000 }{ \displaystyle f(x)=0 }\) when evaluated at \(\color{#000000 }{ \displaystyle x=a,~{\small \tt or}~b }\)
\([\rm part~3]\) \(\color{#000000 }{ \displaystyle f(x)=0 }\) when evaluated at \(\color{#000000 }{ \displaystyle x=a,~b,~{\small \tt or}~c }\)

Answer 3

If something is not making sense, then ask...

Answer 4

NOTE:
This applies to absolutely all numbers \(a, b, c ~....\) (Complex \(\mathbb{C}\), or Real \(\mathbb{R}\))

Answer 5

it makes sense, thank you @SolomonZelman and @AlexandervonHumboldt2

Answer 6

No problem .

Answer 7

Principle to follow here: If a is a root of a given polynomial, then (x-a) is a factor.
Look at the given roots and turn them into factors.
Note that if you have a complex root, you must find the conjugate of that root; both must be turned into factors. Show your work, please.

Answer 8

If one root is -2, what is the corresponding factor?

Answer 9

Mathmale, there is only one complex root. (It's possible for a polynomial to have complex coefficients)