How to prove that polynomial equation $x^6 − 5 x + 2$ has any (at least) one root?











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As the question states, how do I prove that this equation has at least one root?
I tried doing the derivative, but it isn't quite conclusive as it just states the direction of the graph (if it's growing or decreasing), and it doesn't prove it crosses the origin (has one root)
How do I prove it?










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    What is $f(0)$? What is $f(1)$?
    – lulu
    yesterday

















up vote
0
down vote

favorite
1












As the question states, how do I prove that this equation has at least one root?
I tried doing the derivative, but it isn't quite conclusive as it just states the direction of the graph (if it's growing or decreasing), and it doesn't prove it crosses the origin (has one root)
How do I prove it?










share|cite|improve this question









New contributor




j.doe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • 4




    What is $f(0)$? What is $f(1)$?
    – lulu
    yesterday















up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





As the question states, how do I prove that this equation has at least one root?
I tried doing the derivative, but it isn't quite conclusive as it just states the direction of the graph (if it's growing or decreasing), and it doesn't prove it crosses the origin (has one root)
How do I prove it?










share|cite|improve this question









New contributor




j.doe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











As the question states, how do I prove that this equation has at least one root?
I tried doing the derivative, but it isn't quite conclusive as it just states the direction of the graph (if it's growing or decreasing), and it doesn't prove it crosses the origin (has one root)
How do I prove it?







derivatives polynomials






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edited yesterday









PradyumanDixit

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asked yesterday









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  • 4




    What is $f(0)$? What is $f(1)$?
    – lulu
    yesterday
















  • 4




    What is $f(0)$? What is $f(1)$?
    – lulu
    yesterday










4




4




What is $f(0)$? What is $f(1)$?
– lulu
yesterday






What is $f(0)$? What is $f(1)$?
– lulu
yesterday












3 Answers
3






active

oldest

votes

















up vote
2
down vote



accepted










let $f(x) = x^6 - 5x + 2$



$f(1) = -3$



$f(0) = 2$



There is at least one root between $0$ and $1$



$f(2)= 56$



There must also be at least one root between $1$ and $2$






share|cite|improve this answer




























    up vote
    1
    down vote













    In general, to show that $f(x)$ has a root (and as an added bonus find a bound for that root), you can try out different values of $x$. If you can find real numbers $a$ and $b$ with $f(a)<0<f(b)$, then you know that there must be some $c$ in $(a,b)$ with $f(c)=0$. This result is known as the intermediate value theorem, a well known fact in real analysis applying to continuous functions $f:[a,b]tomathbb R$ ($a$ and $b$ are real numbers). Here, just test the values $a=0$ and $b=1$ like other answers have suggested to get your proof.






    share|cite|improve this answer




























      up vote
      0
      down vote













      Your best bet is the use of the intermediate value theorem, assuming you want a calculus-based approach.



      Show that there exists $x_1,x_2$ such that $f(x_1) < 0$ and $f(x_2) > 0$. Then by the intermediate value theorem, as $f$ is clearly continuous, there exists a $c in (x_1,x_2)$ such that $f(c) = 0$ - and thus, $f$ has a root!



      (You can test arbitrary $x_1,x_2$, picking at random if you choose. The intermediate value theorem doesn't tell you how to pick the $x_1,x_2$, just what happens if such points exist.)






      share|cite|improve this answer





















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        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes








        up vote
        2
        down vote



        accepted










        let $f(x) = x^6 - 5x + 2$



        $f(1) = -3$



        $f(0) = 2$



        There is at least one root between $0$ and $1$



        $f(2)= 56$



        There must also be at least one root between $1$ and $2$






        share|cite|improve this answer

























          up vote
          2
          down vote



          accepted










          let $f(x) = x^6 - 5x + 2$



          $f(1) = -3$



          $f(0) = 2$



          There is at least one root between $0$ and $1$



          $f(2)= 56$



          There must also be at least one root between $1$ and $2$






          share|cite|improve this answer























            up vote
            2
            down vote



            accepted







            up vote
            2
            down vote



            accepted






            let $f(x) = x^6 - 5x + 2$



            $f(1) = -3$



            $f(0) = 2$



            There is at least one root between $0$ and $1$



            $f(2)= 56$



            There must also be at least one root between $1$ and $2$






            share|cite|improve this answer












            let $f(x) = x^6 - 5x + 2$



            $f(1) = -3$



            $f(0) = 2$



            There is at least one root between $0$ and $1$



            $f(2)= 56$



            There must also be at least one root between $1$ and $2$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered yesterday









            Doug M

            42.4k31752




            42.4k31752






















                up vote
                1
                down vote













                In general, to show that $f(x)$ has a root (and as an added bonus find a bound for that root), you can try out different values of $x$. If you can find real numbers $a$ and $b$ with $f(a)<0<f(b)$, then you know that there must be some $c$ in $(a,b)$ with $f(c)=0$. This result is known as the intermediate value theorem, a well known fact in real analysis applying to continuous functions $f:[a,b]tomathbb R$ ($a$ and $b$ are real numbers). Here, just test the values $a=0$ and $b=1$ like other answers have suggested to get your proof.






                share|cite|improve this answer

























                  up vote
                  1
                  down vote













                  In general, to show that $f(x)$ has a root (and as an added bonus find a bound for that root), you can try out different values of $x$. If you can find real numbers $a$ and $b$ with $f(a)<0<f(b)$, then you know that there must be some $c$ in $(a,b)$ with $f(c)=0$. This result is known as the intermediate value theorem, a well known fact in real analysis applying to continuous functions $f:[a,b]tomathbb R$ ($a$ and $b$ are real numbers). Here, just test the values $a=0$ and $b=1$ like other answers have suggested to get your proof.






                  share|cite|improve this answer























                    up vote
                    1
                    down vote










                    up vote
                    1
                    down vote









                    In general, to show that $f(x)$ has a root (and as an added bonus find a bound for that root), you can try out different values of $x$. If you can find real numbers $a$ and $b$ with $f(a)<0<f(b)$, then you know that there must be some $c$ in $(a,b)$ with $f(c)=0$. This result is known as the intermediate value theorem, a well known fact in real analysis applying to continuous functions $f:[a,b]tomathbb R$ ($a$ and $b$ are real numbers). Here, just test the values $a=0$ and $b=1$ like other answers have suggested to get your proof.






                    share|cite|improve this answer












                    In general, to show that $f(x)$ has a root (and as an added bonus find a bound for that root), you can try out different values of $x$. If you can find real numbers $a$ and $b$ with $f(a)<0<f(b)$, then you know that there must be some $c$ in $(a,b)$ with $f(c)=0$. This result is known as the intermediate value theorem, a well known fact in real analysis applying to continuous functions $f:[a,b]tomathbb R$ ($a$ and $b$ are real numbers). Here, just test the values $a=0$ and $b=1$ like other answers have suggested to get your proof.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered yesterday









                    YiFan

                    1,3281211




                    1,3281211






















                        up vote
                        0
                        down vote













                        Your best bet is the use of the intermediate value theorem, assuming you want a calculus-based approach.



                        Show that there exists $x_1,x_2$ such that $f(x_1) < 0$ and $f(x_2) > 0$. Then by the intermediate value theorem, as $f$ is clearly continuous, there exists a $c in (x_1,x_2)$ such that $f(c) = 0$ - and thus, $f$ has a root!



                        (You can test arbitrary $x_1,x_2$, picking at random if you choose. The intermediate value theorem doesn't tell you how to pick the $x_1,x_2$, just what happens if such points exist.)






                        share|cite|improve this answer

























                          up vote
                          0
                          down vote













                          Your best bet is the use of the intermediate value theorem, assuming you want a calculus-based approach.



                          Show that there exists $x_1,x_2$ such that $f(x_1) < 0$ and $f(x_2) > 0$. Then by the intermediate value theorem, as $f$ is clearly continuous, there exists a $c in (x_1,x_2)$ such that $f(c) = 0$ - and thus, $f$ has a root!



                          (You can test arbitrary $x_1,x_2$, picking at random if you choose. The intermediate value theorem doesn't tell you how to pick the $x_1,x_2$, just what happens if such points exist.)






                          share|cite|improve this answer























                            up vote
                            0
                            down vote










                            up vote
                            0
                            down vote









                            Your best bet is the use of the intermediate value theorem, assuming you want a calculus-based approach.



                            Show that there exists $x_1,x_2$ such that $f(x_1) < 0$ and $f(x_2) > 0$. Then by the intermediate value theorem, as $f$ is clearly continuous, there exists a $c in (x_1,x_2)$ such that $f(c) = 0$ - and thus, $f$ has a root!



                            (You can test arbitrary $x_1,x_2$, picking at random if you choose. The intermediate value theorem doesn't tell you how to pick the $x_1,x_2$, just what happens if such points exist.)






                            share|cite|improve this answer












                            Your best bet is the use of the intermediate value theorem, assuming you want a calculus-based approach.



                            Show that there exists $x_1,x_2$ such that $f(x_1) < 0$ and $f(x_2) > 0$. Then by the intermediate value theorem, as $f$ is clearly continuous, there exists a $c in (x_1,x_2)$ such that $f(c) = 0$ - and thus, $f$ has a root!



                            (You can test arbitrary $x_1,x_2$, picking at random if you choose. The intermediate value theorem doesn't tell you how to pick the $x_1,x_2$, just what happens if such points exist.)







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered yesterday









                            Eevee Trainer

                            6218




                            6218






















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