Find $p$ if the remainder when $(x^2 + px + 13)$ divided by $(x-2)$ is twice the remainder when it is divided...












2












$begingroup$


As far as I understand, this should be a simultaneous equation, although I'm not really sure where to go beyond that. I made the remainder of $f(-2)=r$ and $f(2)=2r$, and substituted the respective values.



$x^2+px+13$



With $(x-2)$



$f(2)=(2)^2+p(2)+13$



$f(2)=17+2p=2r$



With $(x+2)$



$f(-2)=(-2)^2+p(-2)+13$



$f(-2)=17-2p=r$



I guess I could do a simultaneous here, and I got $17/6$ as my result, but I'm not sure if that's correct.










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$endgroup$








  • 1




    $begingroup$
    looks good to me
    $endgroup$
    – randomgirl
    Dec 17 '18 at 16:20
















2












$begingroup$


As far as I understand, this should be a simultaneous equation, although I'm not really sure where to go beyond that. I made the remainder of $f(-2)=r$ and $f(2)=2r$, and substituted the respective values.



$x^2+px+13$



With $(x-2)$



$f(2)=(2)^2+p(2)+13$



$f(2)=17+2p=2r$



With $(x+2)$



$f(-2)=(-2)^2+p(-2)+13$



$f(-2)=17-2p=r$



I guess I could do a simultaneous here, and I got $17/6$ as my result, but I'm not sure if that's correct.










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    looks good to me
    $endgroup$
    – randomgirl
    Dec 17 '18 at 16:20














2












2








2





$begingroup$


As far as I understand, this should be a simultaneous equation, although I'm not really sure where to go beyond that. I made the remainder of $f(-2)=r$ and $f(2)=2r$, and substituted the respective values.



$x^2+px+13$



With $(x-2)$



$f(2)=(2)^2+p(2)+13$



$f(2)=17+2p=2r$



With $(x+2)$



$f(-2)=(-2)^2+p(-2)+13$



$f(-2)=17-2p=r$



I guess I could do a simultaneous here, and I got $17/6$ as my result, but I'm not sure if that's correct.










share|cite|improve this question











$endgroup$




As far as I understand, this should be a simultaneous equation, although I'm not really sure where to go beyond that. I made the remainder of $f(-2)=r$ and $f(2)=2r$, and substituted the respective values.



$x^2+px+13$



With $(x-2)$



$f(2)=(2)^2+p(2)+13$



$f(2)=17+2p=2r$



With $(x+2)$



$f(-2)=(-2)^2+p(-2)+13$



$f(-2)=17-2p=r$



I guess I could do a simultaneous here, and I got $17/6$ as my result, but I'm not sure if that's correct.







proof-verification polynomials






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edited Dec 17 '18 at 16:08









Tianlalu

3,09421138




3,09421138










asked Dec 17 '18 at 16:04









Roo23Roo23

213




213








  • 1




    $begingroup$
    looks good to me
    $endgroup$
    – randomgirl
    Dec 17 '18 at 16:20














  • 1




    $begingroup$
    looks good to me
    $endgroup$
    – randomgirl
    Dec 17 '18 at 16:20








1




1




$begingroup$
looks good to me
$endgroup$
– randomgirl
Dec 17 '18 at 16:20




$begingroup$
looks good to me
$endgroup$
– randomgirl
Dec 17 '18 at 16:20










2 Answers
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0












$begingroup$

You are making it too hard.



$f(2) = 17+2p,
f(-2) = 17-2p$
.



So you want
$17+2p = 2(17-2p)
=34-4p$

so
$17=6p$
or
$p = 17/6$
as you got.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    HINT:



    $$x^2+px+13=(x+p+2)(x-2)+(17+2p)=(x+p-2)(x-2)+(17-2p)$$



    Via polynomial long division.






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      2 Answers
      2






      active

      oldest

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      2 Answers
      2






      active

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      0












      $begingroup$

      You are making it too hard.



      $f(2) = 17+2p,
      f(-2) = 17-2p$
      .



      So you want
      $17+2p = 2(17-2p)
      =34-4p$

      so
      $17=6p$
      or
      $p = 17/6$
      as you got.






      share|cite|improve this answer









      $endgroup$


















        0












        $begingroup$

        You are making it too hard.



        $f(2) = 17+2p,
        f(-2) = 17-2p$
        .



        So you want
        $17+2p = 2(17-2p)
        =34-4p$

        so
        $17=6p$
        or
        $p = 17/6$
        as you got.






        share|cite|improve this answer









        $endgroup$
















          0












          0








          0





          $begingroup$

          You are making it too hard.



          $f(2) = 17+2p,
          f(-2) = 17-2p$
          .



          So you want
          $17+2p = 2(17-2p)
          =34-4p$

          so
          $17=6p$
          or
          $p = 17/6$
          as you got.






          share|cite|improve this answer









          $endgroup$



          You are making it too hard.



          $f(2) = 17+2p,
          f(-2) = 17-2p$
          .



          So you want
          $17+2p = 2(17-2p)
          =34-4p$

          so
          $17=6p$
          or
          $p = 17/6$
          as you got.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 17 '18 at 18:15









          marty cohenmarty cohen

          74.4k549129




          74.4k549129























              0












              $begingroup$

              HINT:



              $$x^2+px+13=(x+p+2)(x-2)+(17+2p)=(x+p-2)(x-2)+(17-2p)$$



              Via polynomial long division.






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                HINT:



                $$x^2+px+13=(x+p+2)(x-2)+(17+2p)=(x+p-2)(x-2)+(17-2p)$$



                Via polynomial long division.






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  HINT:



                  $$x^2+px+13=(x+p+2)(x-2)+(17+2p)=(x+p-2)(x-2)+(17-2p)$$



                  Via polynomial long division.






                  share|cite|improve this answer









                  $endgroup$



                  HINT:



                  $$x^2+px+13=(x+p+2)(x-2)+(17+2p)=(x+p-2)(x-2)+(17-2p)$$



                  Via polynomial long division.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 17 '18 at 16:21









                  Rhys HughesRhys Hughes

                  7,0251630




                  7,0251630






























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