Question

I have two questions I am struggling with that I have spent waaay too much time on & now it's crunch time. Any assistance would be appreciated.
Form a polynomial f (x) with real coefficients having the given degree and zeros.
Degree​ 4; ​ zeros: -3-2i; -4 multiplicity 2

Form a polynomial​ f(x) with real coefficients having the given degree and zeros.
Degree​ 4; ​ zeros: 6​, multiplicity​ 2; 2i

Answer 1

whats th answer choices

Answer 2

complex zeros always occur in pairs.
other zero is\[-3+2 \iota\]
\[f(x)=\left\{ x-\left( -3-2 \iota \right) \right\}\left\{ x-\left( -3+2i \right) \right\}\left\{ x-\left( -4 \right) \right \}^2\]\[=\left(( x+3)+2 \iota \right)\left(( x+3)-2 \iota \right)\left( x+4 \right)^2\]
first multiply first two parenthesis
\[\left( a+b \right)\left( a-b \right)=a^2-b^2\]
\[\left( a+b \right)^2=a^2+b^2+2ab\]

Answer 3

"Form a polynomial f (x) with real coefficients having the given degree and zeros.
Degree​ 4; ​ zeros: -3-2i; -4 multiplicity 2"

If the zeros are -3+2i and -4, you need to expect that another zero would be -3+2i (which we call the "complex conjugate" of -3-2i.

Note that there is some ambiguity here. Did you mean to say that the zeros are -3 AND 2i AND -4? Or did you mean to say that one zero is the complex -3+2i?

Answer 4

Important: look up "multiplicity." How does this concept apply to the problem at hand?

Answer 5

I didn't mean to say anything, the problem is written exactly how I typed it. Meaning the one zero is is the complex -3+2i

Answer 6

It's not multiple choice. I think I'm messing up when I expand it. I feel like I'm doing it correctly but not getting the correct answer, so I am definitely missing something.

Answer 7

\[f(x)=\left\{ \left( x+3 \right)^2-\left( 2 \iota \right)^2 \right\}(x+4)^2\]
\[=\left\{ \left( x+3 \right)^2-4 \iota ^2 \right\}\left( x+4 \right)^2\]
\[=\left( x^2+9+6x+4 \right)\left( x^2+8x+16 \right)\]

Answer 8

"Meaning the one zero is is the complex -3+2i" Then another zero is -3-2i.

Write the factors corresponding to these 2 zeros.

Answer 9

simplify further

Answer 10

About multiplicity: You might want to refer to the following:

https://www.google.com/search?q=multiplicity&oq=multiplicity&aqs=chrome..69i57j0l5.2480j0j7&sourceid=chrome&ie=UTF-8

If the multiplicity is 2, then one root is duplicated; e. g., if one root is 2, and the multiplicity is 2, then 2 is a root twice and (x-2) is a factor twice.

Answer 11

"Form a polynomial f (x) with real coefficients having the given degree and zeros.
Degree​ 4; ​ zeros: -3-2i; -4 multiplicity 2" => when you've finished, you should have a polynomial whose highest x power is 4: ax^4 + bx^3 + cx^2 + d.

If -3-2i is one root, then -3+2i is another root. If 2 is a root, then you have two factors: (x-2)(x-2). Write out all four factors and then multiply them together. This is what sshayer has been asking you to do.

Answer 12

Will try it later, thanks everyone

Answer 13

correction last term 13*16