A polynomial is exactly divided by$ x+1$, and when it is divided by $3x-1$, the remainder is $4$…...












-1














A polynomial is exactly divided by $x+1$, and when it is divided by $3x-1$, the remainder is $4$. Given that the polynomial gives a remainder $hx+k$ when divided by $3x^2+2x-1$, find $h$ and $k$



I've been having a little bit of trouble with this question because I'm not entirely sure what to do when the polynomial isn't given.






polynomials







share|cite|improve this question
























  • en.wikipedia.org/wiki/Polynomial_remainder_theorem
    – Matko
    2 days ago
















-1














A polynomial is exactly divided by $x+1$, and when it is divided by $3x-1$, the remainder is $4$. Given that the polynomial gives a remainder $hx+k$ when divided by $3x^2+2x-1$, find $h$ and $k$



I've been having a little bit of trouble with this question because I'm not entirely sure what to do when the polynomial isn't given.






polynomials







share|cite|improve this question
























  • en.wikipedia.org/wiki/Polynomial_remainder_theorem
    – Matko
    2 days ago














-1












-1








-1







A polynomial is exactly divided by $x+1$, and when it is divided by $3x-1$, the remainder is $4$. Given that the polynomial gives a remainder $hx+k$ when divided by $3x^2+2x-1$, find $h$ and $k$



I've been having a little bit of trouble with this question because I'm not entirely sure what to do when the polynomial isn't given.






polynomials







share|cite|improve this question















A polynomial is exactly divided by $x+1$, and when it is divided by $3x-1$, the remainder is $4$. Given that the polynomial gives a remainder $hx+k$ when divided by $3x^2+2x-1$, find $h$ and $k$



I've been having a little bit of trouble with this question because I'm not entirely sure what to do when the polynomial isn't given.






polynomials




polynomials






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago









mrtaurho

3,3782932




3,3782932










asked 2 days ago









Roo23

163




163












  • en.wikipedia.org/wiki/Polynomial_remainder_theorem
    – Matko
    2 days ago


















  • en.wikipedia.org/wiki/Polynomial_remainder_theorem
    – Matko
    2 days ago
















en.wikipedia.org/wiki/Polynomial_remainder_theorem
– Matko
2 days ago




en.wikipedia.org/wiki/Polynomial_remainder_theorem
– Matko
2 days ago










2 Answers
2






active

oldest

votes


















2














Givens:



$P(x) = Q_1(x)cdot (x+1)$ [note here that $P(-1) = 0$]



and



$P(x) = Q_2(x)cdot (3x-1) + 4$ [note here that $P(frac 13) = 4$]



Need to determine $h,k$ where:



$P(x) = Q_3(x)cdot (x+1)(3x-1) + hx + k$



$P(-1) = 0 implies -h + k = 0 implies h = k$.



$P(frac 13)= 4 implies frac h3 + k = 4 implies frac {4h}3 = 4 implies h = k = 3$.






share|cite|improve this answer





























    0














    $ overbrace{f!-!4bmod (3x!-!1)(x!+!1)}^{Large {rm factor out}, 3x,-1 mid f-4 rightarrow!!!!!!!} =, (3x!-!1)!!underbrace{overbrace{left[dfrac{color{#c00}f!-!4}{color{#0a0}{3x}!-!1}bmod x!+!1right]}^{Large color{#c00}fbmod x+1 = color{#c00}0 ,rightarrow!!!!!!!!!!!}}_{Large color{#0a0}gbmod x+1, =, color{#0a0}{g(-1)} rightarrow!!!!!!}!! =, (3x!-!1)left[dfrac{ color{#c00}0!-!4}{color{#0a0}{-3}!-!1}right] =, 3x!-!1$






    share|cite|improve this answer























    • We used the mod Distributive Law $ abbmod ac,=,a(vbmod c) $ to factor out $,a=3x−1.,$ To conclude simply add $4$ to the above to obtain $ ,fequiv 3x!+!3pmod{(3x-1)(x+1)} $
      – Bill Dubuque
      2 hours ago













    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',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    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%2f3050235%2fa-polynomial-is-exactly-divided-by-x1-and-when-it-is-divided-by-3x-1-the%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2














    Givens:



    $P(x) = Q_1(x)cdot (x+1)$ [note here that $P(-1) = 0$]



    and



    $P(x) = Q_2(x)cdot (3x-1) + 4$ [note here that $P(frac 13) = 4$]



    Need to determine $h,k$ where:



    $P(x) = Q_3(x)cdot (x+1)(3x-1) + hx + k$



    $P(-1) = 0 implies -h + k = 0 implies h = k$.



    $P(frac 13)= 4 implies frac h3 + k = 4 implies frac {4h}3 = 4 implies h = k = 3$.






    share|cite|improve this answer


























      2














      Givens:



      $P(x) = Q_1(x)cdot (x+1)$ [note here that $P(-1) = 0$]



      and



      $P(x) = Q_2(x)cdot (3x-1) + 4$ [note here that $P(frac 13) = 4$]



      Need to determine $h,k$ where:



      $P(x) = Q_3(x)cdot (x+1)(3x-1) + hx + k$



      $P(-1) = 0 implies -h + k = 0 implies h = k$.



      $P(frac 13)= 4 implies frac h3 + k = 4 implies frac {4h}3 = 4 implies h = k = 3$.






      share|cite|improve this answer
























        2












        2








        2






        Givens:



        $P(x) = Q_1(x)cdot (x+1)$ [note here that $P(-1) = 0$]



        and



        $P(x) = Q_2(x)cdot (3x-1) + 4$ [note here that $P(frac 13) = 4$]



        Need to determine $h,k$ where:



        $P(x) = Q_3(x)cdot (x+1)(3x-1) + hx + k$



        $P(-1) = 0 implies -h + k = 0 implies h = k$.



        $P(frac 13)= 4 implies frac h3 + k = 4 implies frac {4h}3 = 4 implies h = k = 3$.






        share|cite|improve this answer












        Givens:



        $P(x) = Q_1(x)cdot (x+1)$ [note here that $P(-1) = 0$]



        and



        $P(x) = Q_2(x)cdot (3x-1) + 4$ [note here that $P(frac 13) = 4$]



        Need to determine $h,k$ where:



        $P(x) = Q_3(x)cdot (x+1)(3x-1) + hx + k$



        $P(-1) = 0 implies -h + k = 0 implies h = k$.



        $P(frac 13)= 4 implies frac h3 + k = 4 implies frac {4h}3 = 4 implies h = k = 3$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 2 days ago









        Deepak

        16.6k11436




        16.6k11436























            0














            $ overbrace{f!-!4bmod (3x!-!1)(x!+!1)}^{Large {rm factor out}, 3x,-1 mid f-4 rightarrow!!!!!!!} =, (3x!-!1)!!underbrace{overbrace{left[dfrac{color{#c00}f!-!4}{color{#0a0}{3x}!-!1}bmod x!+!1right]}^{Large color{#c00}fbmod x+1 = color{#c00}0 ,rightarrow!!!!!!!!!!!}}_{Large color{#0a0}gbmod x+1, =, color{#0a0}{g(-1)} rightarrow!!!!!!}!! =, (3x!-!1)left[dfrac{ color{#c00}0!-!4}{color{#0a0}{-3}!-!1}right] =, 3x!-!1$






            share|cite|improve this answer























            • We used the mod Distributive Law $ abbmod ac,=,a(vbmod c) $ to factor out $,a=3x−1.,$ To conclude simply add $4$ to the above to obtain $ ,fequiv 3x!+!3pmod{(3x-1)(x+1)} $
              – Bill Dubuque
              2 hours ago


















            0














            $ overbrace{f!-!4bmod (3x!-!1)(x!+!1)}^{Large {rm factor out}, 3x,-1 mid f-4 rightarrow!!!!!!!} =, (3x!-!1)!!underbrace{overbrace{left[dfrac{color{#c00}f!-!4}{color{#0a0}{3x}!-!1}bmod x!+!1right]}^{Large color{#c00}fbmod x+1 = color{#c00}0 ,rightarrow!!!!!!!!!!!}}_{Large color{#0a0}gbmod x+1, =, color{#0a0}{g(-1)} rightarrow!!!!!!}!! =, (3x!-!1)left[dfrac{ color{#c00}0!-!4}{color{#0a0}{-3}!-!1}right] =, 3x!-!1$






            share|cite|improve this answer























            • We used the mod Distributive Law $ abbmod ac,=,a(vbmod c) $ to factor out $,a=3x−1.,$ To conclude simply add $4$ to the above to obtain $ ,fequiv 3x!+!3pmod{(3x-1)(x+1)} $
              – Bill Dubuque
              2 hours ago
















            0












            0








            0






            $ overbrace{f!-!4bmod (3x!-!1)(x!+!1)}^{Large {rm factor out}, 3x,-1 mid f-4 rightarrow!!!!!!!} =, (3x!-!1)!!underbrace{overbrace{left[dfrac{color{#c00}f!-!4}{color{#0a0}{3x}!-!1}bmod x!+!1right]}^{Large color{#c00}fbmod x+1 = color{#c00}0 ,rightarrow!!!!!!!!!!!}}_{Large color{#0a0}gbmod x+1, =, color{#0a0}{g(-1)} rightarrow!!!!!!}!! =, (3x!-!1)left[dfrac{ color{#c00}0!-!4}{color{#0a0}{-3}!-!1}right] =, 3x!-!1$






            share|cite|improve this answer














            $ overbrace{f!-!4bmod (3x!-!1)(x!+!1)}^{Large {rm factor out}, 3x,-1 mid f-4 rightarrow!!!!!!!} =, (3x!-!1)!!underbrace{overbrace{left[dfrac{color{#c00}f!-!4}{color{#0a0}{3x}!-!1}bmod x!+!1right]}^{Large color{#c00}fbmod x+1 = color{#c00}0 ,rightarrow!!!!!!!!!!!}}_{Large color{#0a0}gbmod x+1, =, color{#0a0}{g(-1)} rightarrow!!!!!!}!! =, (3x!-!1)left[dfrac{ color{#c00}0!-!4}{color{#0a0}{-3}!-!1}right] =, 3x!-!1$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited 21 hours ago

























            answered 2 days ago









            Bill Dubuque

            208k29190626




            208k29190626












            • We used the mod Distributive Law $ abbmod ac,=,a(vbmod c) $ to factor out $,a=3x−1.,$ To conclude simply add $4$ to the above to obtain $ ,fequiv 3x!+!3pmod{(3x-1)(x+1)} $
              – Bill Dubuque
              2 hours ago




















            • We used the mod Distributive Law $ abbmod ac,=,a(vbmod c) $ to factor out $,a=3x−1.,$ To conclude simply add $4$ to the above to obtain $ ,fequiv 3x!+!3pmod{(3x-1)(x+1)} $
              – Bill Dubuque
              2 hours ago


















            We used the mod Distributive Law $ abbmod ac,=,a(vbmod c) $ to factor out $,a=3x−1.,$ To conclude simply add $4$ to the above to obtain $ ,fequiv 3x!+!3pmod{(3x-1)(x+1)} $
            – Bill Dubuque
            2 hours ago






            We used the mod Distributive Law $ abbmod ac,=,a(vbmod c) $ to factor out $,a=3x−1.,$ To conclude simply add $4$ to the above to obtain $ ,fequiv 3x!+!3pmod{(3x-1)(x+1)} $
            – Bill Dubuque
            2 hours ago




















            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.





            Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


            Please pay close attention to the following guidance:


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3050235%2fa-polynomial-is-exactly-divided-by-x1-and-when-it-is-divided-by-3x-1-the%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Questions related to Moebius Transform of Characteristic Function of the Primes

            Image
            2 $begingroup$ Consider the function defined in (1) below related to the fundamental prime counting function $pi(x)$ . Note that A143519(n) is not multiplicative. (1) $quad f(x)=sumlimits_{n=1}^{x}A143519(n)$ https://oeis.org/A143519 The following plot illustrates $f(x)$ defined in formula (1) above. Figure (1) : Illustration of $f(x)$ defined in formula (1) The integer zeros of $f(x)$ for $xle 10,000$ are listed in (2) below. (2) $quad$ {1,6,9,12,19,30,79,80,81,116,193,201,287,288,291,668,673,679,680,685,686,1109} The zero crossings of $f(x)$ for $xle 10,000$ where $f(x)$ doesn't settle at zero are listed in (3) below. (3) $quad$ {14,21,33,114,115,118,195,286,290,295,442,445,665,667,670,671,678,682} Question (1) : Does $f(x)$ have a finite number of integer zeros, and if so ...
            Read more

            List of scandals in India

            Image
            List of scandals in India From Wikipedia, the free encyclopedia Jump to navigation Jump to search This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by expanding it with reliably sourced entries. This article has multiple issues. Please help improve it or discuss these issues on the talk page . (Learn how and when to remove these template messages) This article has an unclear citation style . The references used may be made clearer with a different or consistent style of citation and footnoting. ( January 2014 ) (Learn how and when to remove this template message) This list is in reverse order. As a consequence it appears to be slanted towards recent events. Please try to keep recent events in historical perspective and add more content related to non-recent events. ( May 2013 ) (Learn how and when to remove this template message) ...
            Read more

            Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

            Image
            0 1 What exactly does the error message beginning with E: mean? I assume it has to do with the file structure on Ubuntu-on-Windows, but what exactly? The following package was automatically installed and is no longer required: os-prober Use 'apt-get autoremove' to remove it. The following extra packages will be installed: libxslt1.1 The following NEW packages will be installed: libxslt1.1 xmlstarlet 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 435 kB of archives. After this operation, 1023 kB of additional disk space will be used. Get:1 http://archive.ubuntu.com/ubuntu/ trusty/main libxslt1.1 amd64 1.1.28-2build1 [145 kB] Get:2 http://archive.ubuntu.com/ubuntu/ trusty/universe xmlstarlet amd64 1.5.0-1 [290 kB] ...
            Read more