Finding real coefficients of polynomial with complex roots

Multi tool use
Multi tool use

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
3
down vote

favorite












I have been tasked with finding the just real coefficients of a polynomial with the roots:



$z_0=10$



$z_1=3-i$



$z_2=-8+2i$



Writing the polynomial as factors...



$(z-10)(z-3+i)(z+8-2i)$



..obviously does not work since this results in complex coefficients.



Can anyone help me out here?






algebra-precalculus polynomials complex-numbers







share|cite|improve this question























  • “the real coefficients of a polynomial” or “a polynomial with only real coefficients”?
    – Martin R
    8 hours ago










  • Only real coefficients.
    – Boris Grunwald
    8 hours ago














up vote
3
down vote

favorite












I have been tasked with finding the just real coefficients of a polynomial with the roots:



$z_0=10$



$z_1=3-i$



$z_2=-8+2i$



Writing the polynomial as factors...



$(z-10)(z-3+i)(z+8-2i)$



..obviously does not work since this results in complex coefficients.



Can anyone help me out here?






algebra-precalculus polynomials complex-numbers







share|cite|improve this question























  • “the real coefficients of a polynomial” or “a polynomial with only real coefficients”?
    – Martin R
    8 hours ago










  • Only real coefficients.
    – Boris Grunwald
    8 hours ago












up vote
3
down vote

favorite









up vote
3
down vote

favorite











I have been tasked with finding the just real coefficients of a polynomial with the roots:



$z_0=10$



$z_1=3-i$



$z_2=-8+2i$



Writing the polynomial as factors...



$(z-10)(z-3+i)(z+8-2i)$



..obviously does not work since this results in complex coefficients.



Can anyone help me out here?






algebra-precalculus polynomials complex-numbers







share|cite|improve this question















I have been tasked with finding the just real coefficients of a polynomial with the roots:



$z_0=10$



$z_1=3-i$



$z_2=-8+2i$



Writing the polynomial as factors...



$(z-10)(z-3+i)(z+8-2i)$



..obviously does not work since this results in complex coefficients.



Can anyone help me out here?






algebra-precalculus polynomials complex-numbers




algebra-precalculus polynomials complex-numbers






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 8 hours ago









greedoid

31.1k94287




31.1k94287










asked 8 hours ago









Boris Grunwald

877




877











  • “the real coefficients of a polynomial” or “a polynomial with only real coefficients”?
    – Martin R
    8 hours ago










  • Only real coefficients.
    – Boris Grunwald
    8 hours ago
















  • “the real coefficients of a polynomial” or “a polynomial with only real coefficients”?
    – Martin R
    8 hours ago










  • Only real coefficients.
    – Boris Grunwald
    8 hours ago















“the real coefficients of a polynomial” or “a polynomial with only real coefficients”?
– Martin R
8 hours ago




“the real coefficients of a polynomial” or “a polynomial with only real coefficients”?
– Martin R
8 hours ago












Only real coefficients.
– Boris Grunwald
8 hours ago




Only real coefficients.
– Boris Grunwald
8 hours ago










2 Answers
2






active

oldest

votes

















up vote
5
down vote



accepted










There is not enough information, say the degree of a given polynomial.



Anyway, since it has (only) real coeficients if $a+bi$ is one root, then $a-bi$ is also a root of the polynomial.



So this polynomial is divisible with:



$$q(z)=(z-10)colorred(z-3+i)(z-3-i)colorblue(z+8-2i)(z+8+2i) $$
$$= (z-10)(z^2-6z+10)(z^2+16z+68)$$






share|cite|improve this answer






















  • Thanks, I think this makes sense.
    – Boris Grunwald
    8 hours ago










  • And sorry, it was the smallest possible degree
    – Boris Grunwald
    8 hours ago

















up vote
3
down vote













Polynomial with only real coefficients is
$$( z-10) , ( z-2i+8),(z+2 i+8) , ( z-i-3) , ( z+i-3)\=
z^5-118 z^3-68 z^2+3160 z-6800
$$






share|cite|improve this answer




















    Your Answer




    StackExchange.ifUsing("editor", function ()
    return StackExchange.using("mathjaxEditing", function ()
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    );
    );
    , "mathjax-editing");

    StackExchange.ready(function()
    var channelOptions =
    tags: "".split(" "),
    id: "69"
    ;
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function()
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled)
    StackExchange.using("snippets", function()
    createEditor();
    );

    else
    createEditor();

    );

    function createEditor()
    StackExchange.prepareEditor(
    heartbeatType: 'answer',
    convertImagesToLinks: true,
    noModals: false,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    );



    );













     

    draft saved


    draft discarded


















    StackExchange.ready(
    function ()
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2949062%2ffinding-real-coefficients-of-polynomial-with-complex-roots%23new-answer', 'question_page');

    );

    Post as a guest

















    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    5
    down vote



    accepted










    There is not enough information, say the degree of a given polynomial.



    Anyway, since it has (only) real coeficients if $a+bi$ is one root, then $a-bi$ is also a root of the polynomial.



    So this polynomial is divisible with:



    $$q(z)=(z-10)colorred(z-3+i)(z-3-i)colorblue(z+8-2i)(z+8+2i) $$
    $$= (z-10)(z^2-6z+10)(z^2+16z+68)$$






    share|cite|improve this answer






















    • Thanks, I think this makes sense.
      – Boris Grunwald
      8 hours ago










    • And sorry, it was the smallest possible degree
      – Boris Grunwald
      8 hours ago














    up vote
    5
    down vote



    accepted










    There is not enough information, say the degree of a given polynomial.



    Anyway, since it has (only) real coeficients if $a+bi$ is one root, then $a-bi$ is also a root of the polynomial.



    So this polynomial is divisible with:



    $$q(z)=(z-10)colorred(z-3+i)(z-3-i)colorblue(z+8-2i)(z+8+2i) $$
    $$= (z-10)(z^2-6z+10)(z^2+16z+68)$$






    share|cite|improve this answer






















    • Thanks, I think this makes sense.
      – Boris Grunwald
      8 hours ago










    • And sorry, it was the smallest possible degree
      – Boris Grunwald
      8 hours ago












    up vote
    5
    down vote



    accepted







    up vote
    5
    down vote



    accepted






    There is not enough information, say the degree of a given polynomial.



    Anyway, since it has (only) real coeficients if $a+bi$ is one root, then $a-bi$ is also a root of the polynomial.



    So this polynomial is divisible with:



    $$q(z)=(z-10)colorred(z-3+i)(z-3-i)colorblue(z+8-2i)(z+8+2i) $$
    $$= (z-10)(z^2-6z+10)(z^2+16z+68)$$






    share|cite|improve this answer














    There is not enough information, say the degree of a given polynomial.



    Anyway, since it has (only) real coeficients if $a+bi$ is one root, then $a-bi$ is also a root of the polynomial.



    So this polynomial is divisible with:



    $$q(z)=(z-10)colorred(z-3+i)(z-3-i)colorblue(z+8-2i)(z+8+2i) $$
    $$= (z-10)(z^2-6z+10)(z^2+16z+68)$$







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited 8 hours ago

























    answered 8 hours ago









    greedoid

    31.1k94287




    31.1k94287











    • Thanks, I think this makes sense.
      – Boris Grunwald
      8 hours ago










    • And sorry, it was the smallest possible degree
      – Boris Grunwald
      8 hours ago
















    • Thanks, I think this makes sense.
      – Boris Grunwald
      8 hours ago










    • And sorry, it was the smallest possible degree
      – Boris Grunwald
      8 hours ago















    Thanks, I think this makes sense.
    – Boris Grunwald
    8 hours ago




    Thanks, I think this makes sense.
    – Boris Grunwald
    8 hours ago












    And sorry, it was the smallest possible degree
    – Boris Grunwald
    8 hours ago




    And sorry, it was the smallest possible degree
    – Boris Grunwald
    8 hours ago










    up vote
    3
    down vote













    Polynomial with only real coefficients is
    $$( z-10) , ( z-2i+8),(z+2 i+8) , ( z-i-3) , ( z+i-3)\=
    z^5-118 z^3-68 z^2+3160 z-6800
    $$






    share|cite|improve this answer
























      up vote
      3
      down vote













      Polynomial with only real coefficients is
      $$( z-10) , ( z-2i+8),(z+2 i+8) , ( z-i-3) , ( z+i-3)\=
      z^5-118 z^3-68 z^2+3160 z-6800
      $$






      share|cite|improve this answer






















        up vote
        3
        down vote










        up vote
        3
        down vote









        Polynomial with only real coefficients is
        $$( z-10) , ( z-2i+8),(z+2 i+8) , ( z-i-3) , ( z+i-3)\=
        z^5-118 z^3-68 z^2+3160 z-6800
        $$






        share|cite|improve this answer












        Polynomial with only real coefficients is
        $$( z-10) , ( z-2i+8),(z+2 i+8) , ( z-i-3) , ( z+i-3)\=
        z^5-118 z^3-68 z^2+3160 z-6800
        $$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 8 hours ago









        Aleksas Domarkas

        5154




        5154



























             

            draft saved


            draft discarded















































             


            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2949062%2ffinding-real-coefficients-of-polynomial-with-complex-roots%23new-answer', 'question_page');

            );

            Post as a guest
































































            T,p9v7b 0Cjs3x50zEUYPPk3PIjQ6,1Jsrg0U,L6S,D76bBBvwIh uFjACf0ppRKc4W7ksJKvzoC,EU
            -
            4W,ZVb4EXL,at3JxwQt6S

            Popular posts from this blog

            How to check contact read email or not when send email to Individual?

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 1 down vote favorite I'm using WordPress 4.9.8, CiviCRM to 5.5.1, I usually send email to contact by Search> Find contacts View contact details Action> Send email Send email ok, Contact received mail ok like picture But status only Email sent though contact read email or not. So, can CiviCRM can change status to Email read when contact read email? wordpress email share | improve this question asked Sep 26 at 0:12 ToanLuong 49 9 add a comment  |  up vote 1 down vote favorite I'm using WordPress 4.9.8, CiviCRM to 5.5.1, I usually send email to contact by Search> Find contacts View contact details Action> Send email Send email ok, Contact received mail ok like picture But status only Email sent though contact read email or not. So, can CiviCRM can change status to Email read when contact read email? wordpress email share | improve this questi...
            Read more

            How many registers does an x86_64 CPU actually have?

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite I am currently learning reverse engineering and am studying the flags register. I had in my mind that rflags was just another name for one of the 16 general purpose registers, for example rax or rbx . But it looks like rflags is actually an additional register. So that makes 17 registers in total... how many more could there be? I have spent at least an hour on this and found numerous different answers. The best answer so far is this, which says that there are 40 registers in total. 16 General Purpose Registers 2 Status Registers 6 Code Segment Registers 16 SSE Registers 8 FPU/MMX Registers But if I add that up, I get 48. Could anybody provide an official answer on how many registers an x86_64 CPU has (e.g. an Intel i7). Additionally, I have seen references to 'hardware' and 'architectural' registers. What are those registers and how many are there? register x86-64 share | improve this...
            Read more

            Displaying single band from multi-band raster using QGIS

            Image
            Clash Royale CLAN TAG #URR8PPP 1 How can I extract a single band from multi-band raster in QGIS? I have an remote sensed image which has 6 bands (including NDVI band), I want to display each band separately, but have no idea how to do. I have seen some questions similar here but none worked for me. The original image (has 6 bands) is: I want to display the band 6 which should be like this: But I tried gdal_translate, and couldn't get the correct result. What I have got is: qgis raster multi-band share | improve this question edited Mar 5 at 0:53 Summer asked Mar 4 at 6:42 Summer Summer 23 6 Is this any help gis.stackexchange.com/questions/220658/… ? if not gis.stackexchange.com/questions/62133/… might help. – Michael Stimson Mar 4 at 6:46 Thanks for answering but when I used gdal_translate, qgis showed that 'Error 4: Kayena.tif: No such file or directory". Would you know how to fi...
            Read more