Complex polynomial

[Timcago]

What is the lowestWhat is the lowest possible degree of the polynomial with the following zeros: 1-i,2i,4+(the square root of->)3,-1? Find a polynomial with those zeros. Put your answer in expanded form.

My progress so far:

If there is a 1-i, there must also be a 1+i, because of how you solve it via the quadratic formula.

If there is a 2i, there must also be a -2i?

If there is a 4+(the square root of->)3, there must also be a 4-(the square root of->)3?

-1 does not have to have a reciprical

Therefore,

(x-(1-i))(x-(1+i))(x-2i)(x+2i)(x-(4+(the square root of->)3))(x-(4-(the square root of->)3))(x+1)

expanded form: x^7-9x^6+25x^5-47x^4+68x^3-18x^2-64x+104

Is that correct or was a wrong somewhere and its possible to have a lesser degree?

 

[pka]

Why do you need this factor \(\displaystyle \left( {x - \left[ {4 - \sqrt 3 } \right]} \right)\)?

 

[Timcago]

Why do you need this factor \(\displaystyle \left( {x - \left[ {4 - \sqrt 3 } \right]} \right)\)?

How else would it be possible to get the root of a number without solving using the quadratic forumula. When you use the quadratic formula you have a plus or minus sign. If you have \(\displaystyle \left( {x - \left[ {4 + \sqrt 3 } \right]} \right)\) wouldnt you also have \(\displaystyle \left( {x - \left[ {4 - \sqrt 3 } \right]} \right)\)

 

[pka]

Here is the answer:
\(\displaystyle \L
\begin{array}{l}
\left( {x - \left[ {1 - i} \right]} \right)\left( {x - \left[ {1 + i} \right]} \right)\left( {x - \left[ {2i} \right]} \right)\left( {x - \left[ { - 2i} \right]} \right)\left( {x - \left[ {4 + \sqrt 3 } \right]} \right)\left( {x - \left[ { - 1} \right]} \right) \\
= x^6 - \left( {5 + \sqrt 3 } \right)x^5 + \left( {8 + \sqrt 3 } \right)x^4 - \left( {18 + 4\sqrt 3 } \right)x^3 + \left( {8 + 2\sqrt 3 } \right)x^2 + 8x - 32 - 8\sqrt 3 \\
\end{array}\)

 

[tkhunny]

What is the lowest possible degree of the polynomial with the following zeros

Are you SURE the problem statement doesn't require Real Coefficients? If it doesn't, you are way off and your assumptions are incorrect. Go check.

 

[Timcago]

What is the lowest possible degree of the polynomial with the following zeros

Are you SURE the problem statement doesn't require Real Coefficients? If it doesn't, you are way off and your assumptions are incorrect. Go check.

I typed the problem exatly as it was. Anyways, we are learning about how to play with complex polynomials, so non real coeficients seem fine.

Whos solution was right tkhunny, mine or pka's?

I am still unsure whether 4-(the square root of->)3 must also have its reciprical as a zero

 

[tkhunny]

If it doesn't demand Real Coefficients, then it was wrong every time you assumed "there must also be". That just isn't the case.

Oh, and stop saying "Reciprocal". You mean "Complex Conjugate".

 

[pka]

Go to your textbook. Lookup the definition of polynomial.
See if it says anything about the nature of coefficients of polynomials.

Usually I assume that the term polynomial implies real coefficients.
This is particularly true for pre-college mathematics.

However, in some basic algebra courses coefficients of polynomials means rational numbers.

If the textbook is allowing complex coefficients then Tkh is correct.
If the textbook is requiring only real coefficients then I am correct.
On the other hand, if the textbook requires rational coefficients then you are correct.