Simplify third degree polynomial equations.












1












$begingroup$


Given an equation:




$6x^3 - 13x^2 + 8x + 3 = 0$




Broken down to get one form




$(x + {3over2})$




How can you divide the prior equation to know it will simplify to




$(x + {3over2})(6x^2 + 4x + 2) = 0$











share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    One way is: "long division"
    $endgroup$
    – coffeemath
    Dec 8 '18 at 11:23










  • $begingroup$
    There is one real solution and is negative, but it is not -3/2.
    $endgroup$
    – user376343
    Dec 8 '18 at 11:42
















1












$begingroup$


Given an equation:




$6x^3 - 13x^2 + 8x + 3 = 0$




Broken down to get one form




$(x + {3over2})$




How can you divide the prior equation to know it will simplify to




$(x + {3over2})(6x^2 + 4x + 2) = 0$











share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    One way is: "long division"
    $endgroup$
    – coffeemath
    Dec 8 '18 at 11:23










  • $begingroup$
    There is one real solution and is negative, but it is not -3/2.
    $endgroup$
    – user376343
    Dec 8 '18 at 11:42














1












1








1


0



$begingroup$


Given an equation:




$6x^3 - 13x^2 + 8x + 3 = 0$




Broken down to get one form




$(x + {3over2})$




How can you divide the prior equation to know it will simplify to




$(x + {3over2})(6x^2 + 4x + 2) = 0$











share|cite|improve this question











$endgroup$




Given an equation:




$6x^3 - 13x^2 + 8x + 3 = 0$




Broken down to get one form




$(x + {3over2})$




How can you divide the prior equation to know it will simplify to




$(x + {3over2})(6x^2 + 4x + 2) = 0$








algebra-precalculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 8 '18 at 11:25









user587054

51911




51911










asked Dec 8 '18 at 11:20









Samuel M.Samuel M.

1204




1204








  • 1




    $begingroup$
    One way is: "long division"
    $endgroup$
    – coffeemath
    Dec 8 '18 at 11:23










  • $begingroup$
    There is one real solution and is negative, but it is not -3/2.
    $endgroup$
    – user376343
    Dec 8 '18 at 11:42














  • 1




    $begingroup$
    One way is: "long division"
    $endgroup$
    – coffeemath
    Dec 8 '18 at 11:23










  • $begingroup$
    There is one real solution and is negative, but it is not -3/2.
    $endgroup$
    – user376343
    Dec 8 '18 at 11:42








1




1




$begingroup$
One way is: "long division"
$endgroup$
– coffeemath
Dec 8 '18 at 11:23




$begingroup$
One way is: "long division"
$endgroup$
– coffeemath
Dec 8 '18 at 11:23












$begingroup$
There is one real solution and is negative, but it is not -3/2.
$endgroup$
– user376343
Dec 8 '18 at 11:42




$begingroup$
There is one real solution and is negative, but it is not -3/2.
$endgroup$
– user376343
Dec 8 '18 at 11:42










2 Answers
2






active

oldest

votes


















1












$begingroup$

It looks like you have typo. If you take the polynomial $$p(x)=6 x^3 + 13 x^2 + 8 x + 3,$$ then $-3/2$ is a root.



This guarantees that $x+3/2$ is a factor of $p$. This follows easily from the Division Algorithm: if you write $$p=(x+3/2) q(x)+r(x),$$ with $deg r<deg (x+3/2)=1$, then $r$ is constant; from $p(3/2)=0$ it follows that $r=0$. So now you want to find $q$. You can do it by long division, or simply writing $q(x)=ax^2+bx+c$, and then you have
begin{align}
6 x^3 + 13 x^2 + 8 x + 3&=(x+3/2)(ax^2+bx+c)\
&=ax^3+(3a/2+b)x^2+(c+3b/2)x+3c/2.
end{align}

So $3c/2=3$, and $c=2$. Then $2+3b/2=8$, and $b=4$; finally, $3a/2+4=13$, and $a=6$. Thus
$$
q(x)=6x^2+4x+2.
$$






share|cite|improve this answer











$endgroup$





















    1












    $begingroup$

    Following @MartinArgerami's discovery that there was probably a typo.



    if we have a degree of three polynomial we can use the Rational Zero Theorem as such:




    Ration Zero Theorem: For $f(x) = a_nx^n + a_{n-1}x^{n-1}+...+a_2x^2 + a_1x + a_0$, every rational zero of $f(x)$, $frac{p}{q}$ has $p$ as a factor of $a_0$ and $q$ as a factor of $a_n$




    We have the polynomial $f(x) = 6x^3 + 13x + 8x + 3$



    p) Factors of the constant $a_0 = 3$ : $pm 1$, $pm3$



    q) Factors of the coefficient $a_n = 6$ : $pm1$, $pm2$, $pm3$, $pm6$



    $frac{p}{q_1}$ : $pm1$, $pm3$



    $frac{p}{q_2}$ : $pmfrac{1}{2}$, $pmfrac{3}{2}$



    $frac{p}{q_3}$ : $pmfrac{1}{3}$, $pmfrac{3}{3}$



    $frac{p}{q_4}$ : $pmfrac{1}{6}$, $pmfrac{3}{6}$



    We can see that some of these possible zeros are repeated so we can cross some of them out. Our final possible zero list is the following:



    $x = pmfrac{1}{6}$, $pmfrac{1}{3}$, $pmfrac{1}{2}$, $pm1$, $pmfrac{3}{2}$, $pm3$



    Using synthetic division we try these possible factors until we find one. Let's try $-3$.
    When we try $-3$, we are saying, let us see if $(x+3)$ is a factor of $f(x)$



    $begin{array}{c|rrr}&6&13&8&3\-3&&-18&15&-69\hline\&6&-5&23&-66\end{array}$



    We have a remainder of $-66$ so $-3$ is not a factor. We are looking for a remainder of $0$. For the sake of time, let us try $frac{-3}{2}$ since you have found already that it is a factor.



    $begin{array}{c|rrr}&6&13&8&3\-frac{3}{2}&&-9&-6&-3\hline\&6&4&2&0\end{array}$



    We now have a $0$ remainder and $-frac{3}{2}$ is a zero.

    Thus, $f(x)$ can be written like this:



    $f(x) = divisor(x) times quotient(x) + remainder(x) = d(x)q(x) + r(x) \
    = (x+frac{3}{2})(6x^2+4x+2)$



    you can further factor the quotient to find the remaining zeros of $f(x)$






    share|cite|improve this answer











    $endgroup$













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      2 Answers
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      active

      oldest

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      2 Answers
      2






      active

      oldest

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      active

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      active

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      1












      $begingroup$

      It looks like you have typo. If you take the polynomial $$p(x)=6 x^3 + 13 x^2 + 8 x + 3,$$ then $-3/2$ is a root.



      This guarantees that $x+3/2$ is a factor of $p$. This follows easily from the Division Algorithm: if you write $$p=(x+3/2) q(x)+r(x),$$ with $deg r<deg (x+3/2)=1$, then $r$ is constant; from $p(3/2)=0$ it follows that $r=0$. So now you want to find $q$. You can do it by long division, or simply writing $q(x)=ax^2+bx+c$, and then you have
      begin{align}
      6 x^3 + 13 x^2 + 8 x + 3&=(x+3/2)(ax^2+bx+c)\
      &=ax^3+(3a/2+b)x^2+(c+3b/2)x+3c/2.
      end{align}

      So $3c/2=3$, and $c=2$. Then $2+3b/2=8$, and $b=4$; finally, $3a/2+4=13$, and $a=6$. Thus
      $$
      q(x)=6x^2+4x+2.
      $$






      share|cite|improve this answer











      $endgroup$


















        1












        $begingroup$

        It looks like you have typo. If you take the polynomial $$p(x)=6 x^3 + 13 x^2 + 8 x + 3,$$ then $-3/2$ is a root.



        This guarantees that $x+3/2$ is a factor of $p$. This follows easily from the Division Algorithm: if you write $$p=(x+3/2) q(x)+r(x),$$ with $deg r<deg (x+3/2)=1$, then $r$ is constant; from $p(3/2)=0$ it follows that $r=0$. So now you want to find $q$. You can do it by long division, or simply writing $q(x)=ax^2+bx+c$, and then you have
        begin{align}
        6 x^3 + 13 x^2 + 8 x + 3&=(x+3/2)(ax^2+bx+c)\
        &=ax^3+(3a/2+b)x^2+(c+3b/2)x+3c/2.
        end{align}

        So $3c/2=3$, and $c=2$. Then $2+3b/2=8$, and $b=4$; finally, $3a/2+4=13$, and $a=6$. Thus
        $$
        q(x)=6x^2+4x+2.
        $$






        share|cite|improve this answer











        $endgroup$
















          1












          1








          1





          $begingroup$

          It looks like you have typo. If you take the polynomial $$p(x)=6 x^3 + 13 x^2 + 8 x + 3,$$ then $-3/2$ is a root.



          This guarantees that $x+3/2$ is a factor of $p$. This follows easily from the Division Algorithm: if you write $$p=(x+3/2) q(x)+r(x),$$ with $deg r<deg (x+3/2)=1$, then $r$ is constant; from $p(3/2)=0$ it follows that $r=0$. So now you want to find $q$. You can do it by long division, or simply writing $q(x)=ax^2+bx+c$, and then you have
          begin{align}
          6 x^3 + 13 x^2 + 8 x + 3&=(x+3/2)(ax^2+bx+c)\
          &=ax^3+(3a/2+b)x^2+(c+3b/2)x+3c/2.
          end{align}

          So $3c/2=3$, and $c=2$. Then $2+3b/2=8$, and $b=4$; finally, $3a/2+4=13$, and $a=6$. Thus
          $$
          q(x)=6x^2+4x+2.
          $$






          share|cite|improve this answer











          $endgroup$



          It looks like you have typo. If you take the polynomial $$p(x)=6 x^3 + 13 x^2 + 8 x + 3,$$ then $-3/2$ is a root.



          This guarantees that $x+3/2$ is a factor of $p$. This follows easily from the Division Algorithm: if you write $$p=(x+3/2) q(x)+r(x),$$ with $deg r<deg (x+3/2)=1$, then $r$ is constant; from $p(3/2)=0$ it follows that $r=0$. So now you want to find $q$. You can do it by long division, or simply writing $q(x)=ax^2+bx+c$, and then you have
          begin{align}
          6 x^3 + 13 x^2 + 8 x + 3&=(x+3/2)(ax^2+bx+c)\
          &=ax^3+(3a/2+b)x^2+(c+3b/2)x+3c/2.
          end{align}

          So $3c/2=3$, and $c=2$. Then $2+3b/2=8$, and $b=4$; finally, $3a/2+4=13$, and $a=6$. Thus
          $$
          q(x)=6x^2+4x+2.
          $$







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 10 '18 at 18:12

























          answered Dec 9 '18 at 1:28









          Martin ArgeramiMartin Argerami

          127k1182182




          127k1182182























              1












              $begingroup$

              Following @MartinArgerami's discovery that there was probably a typo.



              if we have a degree of three polynomial we can use the Rational Zero Theorem as such:




              Ration Zero Theorem: For $f(x) = a_nx^n + a_{n-1}x^{n-1}+...+a_2x^2 + a_1x + a_0$, every rational zero of $f(x)$, $frac{p}{q}$ has $p$ as a factor of $a_0$ and $q$ as a factor of $a_n$




              We have the polynomial $f(x) = 6x^3 + 13x + 8x + 3$



              p) Factors of the constant $a_0 = 3$ : $pm 1$, $pm3$



              q) Factors of the coefficient $a_n = 6$ : $pm1$, $pm2$, $pm3$, $pm6$



              $frac{p}{q_1}$ : $pm1$, $pm3$



              $frac{p}{q_2}$ : $pmfrac{1}{2}$, $pmfrac{3}{2}$



              $frac{p}{q_3}$ : $pmfrac{1}{3}$, $pmfrac{3}{3}$



              $frac{p}{q_4}$ : $pmfrac{1}{6}$, $pmfrac{3}{6}$



              We can see that some of these possible zeros are repeated so we can cross some of them out. Our final possible zero list is the following:



              $x = pmfrac{1}{6}$, $pmfrac{1}{3}$, $pmfrac{1}{2}$, $pm1$, $pmfrac{3}{2}$, $pm3$



              Using synthetic division we try these possible factors until we find one. Let's try $-3$.
              When we try $-3$, we are saying, let us see if $(x+3)$ is a factor of $f(x)$



              $begin{array}{c|rrr}&6&13&8&3\-3&&-18&15&-69\hline\&6&-5&23&-66\end{array}$



              We have a remainder of $-66$ so $-3$ is not a factor. We are looking for a remainder of $0$. For the sake of time, let us try $frac{-3}{2}$ since you have found already that it is a factor.



              $begin{array}{c|rrr}&6&13&8&3\-frac{3}{2}&&-9&-6&-3\hline\&6&4&2&0\end{array}$



              We now have a $0$ remainder and $-frac{3}{2}$ is a zero.

              Thus, $f(x)$ can be written like this:



              $f(x) = divisor(x) times quotient(x) + remainder(x) = d(x)q(x) + r(x) \
              = (x+frac{3}{2})(6x^2+4x+2)$



              you can further factor the quotient to find the remaining zeros of $f(x)$






              share|cite|improve this answer











              $endgroup$


















                1












                $begingroup$

                Following @MartinArgerami's discovery that there was probably a typo.



                if we have a degree of three polynomial we can use the Rational Zero Theorem as such:




                Ration Zero Theorem: For $f(x) = a_nx^n + a_{n-1}x^{n-1}+...+a_2x^2 + a_1x + a_0$, every rational zero of $f(x)$, $frac{p}{q}$ has $p$ as a factor of $a_0$ and $q$ as a factor of $a_n$




                We have the polynomial $f(x) = 6x^3 + 13x + 8x + 3$



                p) Factors of the constant $a_0 = 3$ : $pm 1$, $pm3$



                q) Factors of the coefficient $a_n = 6$ : $pm1$, $pm2$, $pm3$, $pm6$



                $frac{p}{q_1}$ : $pm1$, $pm3$



                $frac{p}{q_2}$ : $pmfrac{1}{2}$, $pmfrac{3}{2}$



                $frac{p}{q_3}$ : $pmfrac{1}{3}$, $pmfrac{3}{3}$



                $frac{p}{q_4}$ : $pmfrac{1}{6}$, $pmfrac{3}{6}$



                We can see that some of these possible zeros are repeated so we can cross some of them out. Our final possible zero list is the following:



                $x = pmfrac{1}{6}$, $pmfrac{1}{3}$, $pmfrac{1}{2}$, $pm1$, $pmfrac{3}{2}$, $pm3$



                Using synthetic division we try these possible factors until we find one. Let's try $-3$.
                When we try $-3$, we are saying, let us see if $(x+3)$ is a factor of $f(x)$



                $begin{array}{c|rrr}&6&13&8&3\-3&&-18&15&-69\hline\&6&-5&23&-66\end{array}$



                We have a remainder of $-66$ so $-3$ is not a factor. We are looking for a remainder of $0$. For the sake of time, let us try $frac{-3}{2}$ since you have found already that it is a factor.



                $begin{array}{c|rrr}&6&13&8&3\-frac{3}{2}&&-9&-6&-3\hline\&6&4&2&0\end{array}$



                We now have a $0$ remainder and $-frac{3}{2}$ is a zero.

                Thus, $f(x)$ can be written like this:



                $f(x) = divisor(x) times quotient(x) + remainder(x) = d(x)q(x) + r(x) \
                = (x+frac{3}{2})(6x^2+4x+2)$



                you can further factor the quotient to find the remaining zeros of $f(x)$






                share|cite|improve this answer











                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  Following @MartinArgerami's discovery that there was probably a typo.



                  if we have a degree of three polynomial we can use the Rational Zero Theorem as such:




                  Ration Zero Theorem: For $f(x) = a_nx^n + a_{n-1}x^{n-1}+...+a_2x^2 + a_1x + a_0$, every rational zero of $f(x)$, $frac{p}{q}$ has $p$ as a factor of $a_0$ and $q$ as a factor of $a_n$




                  We have the polynomial $f(x) = 6x^3 + 13x + 8x + 3$



                  p) Factors of the constant $a_0 = 3$ : $pm 1$, $pm3$



                  q) Factors of the coefficient $a_n = 6$ : $pm1$, $pm2$, $pm3$, $pm6$



                  $frac{p}{q_1}$ : $pm1$, $pm3$



                  $frac{p}{q_2}$ : $pmfrac{1}{2}$, $pmfrac{3}{2}$



                  $frac{p}{q_3}$ : $pmfrac{1}{3}$, $pmfrac{3}{3}$



                  $frac{p}{q_4}$ : $pmfrac{1}{6}$, $pmfrac{3}{6}$



                  We can see that some of these possible zeros are repeated so we can cross some of them out. Our final possible zero list is the following:



                  $x = pmfrac{1}{6}$, $pmfrac{1}{3}$, $pmfrac{1}{2}$, $pm1$, $pmfrac{3}{2}$, $pm3$



                  Using synthetic division we try these possible factors until we find one. Let's try $-3$.
                  When we try $-3$, we are saying, let us see if $(x+3)$ is a factor of $f(x)$



                  $begin{array}{c|rrr}&6&13&8&3\-3&&-18&15&-69\hline\&6&-5&23&-66\end{array}$



                  We have a remainder of $-66$ so $-3$ is not a factor. We are looking for a remainder of $0$. For the sake of time, let us try $frac{-3}{2}$ since you have found already that it is a factor.



                  $begin{array}{c|rrr}&6&13&8&3\-frac{3}{2}&&-9&-6&-3\hline\&6&4&2&0\end{array}$



                  We now have a $0$ remainder and $-frac{3}{2}$ is a zero.

                  Thus, $f(x)$ can be written like this:



                  $f(x) = divisor(x) times quotient(x) + remainder(x) = d(x)q(x) + r(x) \
                  = (x+frac{3}{2})(6x^2+4x+2)$



                  you can further factor the quotient to find the remaining zeros of $f(x)$






                  share|cite|improve this answer











                  $endgroup$



                  Following @MartinArgerami's discovery that there was probably a typo.



                  if we have a degree of three polynomial we can use the Rational Zero Theorem as such:




                  Ration Zero Theorem: For $f(x) = a_nx^n + a_{n-1}x^{n-1}+...+a_2x^2 + a_1x + a_0$, every rational zero of $f(x)$, $frac{p}{q}$ has $p$ as a factor of $a_0$ and $q$ as a factor of $a_n$




                  We have the polynomial $f(x) = 6x^3 + 13x + 8x + 3$



                  p) Factors of the constant $a_0 = 3$ : $pm 1$, $pm3$



                  q) Factors of the coefficient $a_n = 6$ : $pm1$, $pm2$, $pm3$, $pm6$



                  $frac{p}{q_1}$ : $pm1$, $pm3$



                  $frac{p}{q_2}$ : $pmfrac{1}{2}$, $pmfrac{3}{2}$



                  $frac{p}{q_3}$ : $pmfrac{1}{3}$, $pmfrac{3}{3}$



                  $frac{p}{q_4}$ : $pmfrac{1}{6}$, $pmfrac{3}{6}$



                  We can see that some of these possible zeros are repeated so we can cross some of them out. Our final possible zero list is the following:



                  $x = pmfrac{1}{6}$, $pmfrac{1}{3}$, $pmfrac{1}{2}$, $pm1$, $pmfrac{3}{2}$, $pm3$



                  Using synthetic division we try these possible factors until we find one. Let's try $-3$.
                  When we try $-3$, we are saying, let us see if $(x+3)$ is a factor of $f(x)$



                  $begin{array}{c|rrr}&6&13&8&3\-3&&-18&15&-69\hline\&6&-5&23&-66\end{array}$



                  We have a remainder of $-66$ so $-3$ is not a factor. We are looking for a remainder of $0$. For the sake of time, let us try $frac{-3}{2}$ since you have found already that it is a factor.



                  $begin{array}{c|rrr}&6&13&8&3\-frac{3}{2}&&-9&-6&-3\hline\&6&4&2&0\end{array}$



                  We now have a $0$ remainder and $-frac{3}{2}$ is a zero.

                  Thus, $f(x)$ can be written like this:



                  $f(x) = divisor(x) times quotient(x) + remainder(x) = d(x)q(x) + r(x) \
                  = (x+frac{3}{2})(6x^2+4x+2)$



                  you can further factor the quotient to find the remaining zeros of $f(x)$







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Dec 10 '18 at 16:28

























                  answered Dec 9 '18 at 2:32









                  BucephalusBucephalus

                  665518




                  665518






























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