Rational Roots Test / Theorem

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  • Опубліковано 17 лис 2024
  • Rational Roots Test - In this video I talk about how you can find all of the possible rational roots of a polynomial.
    Note that to find the rational roots of a polynomial we simply look at all possible combinations of factors of the constant divided by factors of the leading coefficient.
    Note that if there is not a constant, all terms involve some power of x. In this case FIRST factor out the smallest power of x and then use the rational roots test on the remaining factored polynomial.
    Once you have all of the possible rational roots it is just a matter of substituting each candidate into the original polynomial to see if any of these values give zero after simplifying.
    #maths #patrickjmt #rationalroots #factorial #polynomials #zeroes #zeroesofpolynomial

КОМЕНТАРІ • 468

  • @arav4045
    @arav4045 4 роки тому +392

    I was three year olds when this video was made. Now this same video is helping me 11 years later

  • @pizzafreak1175
    @pizzafreak1175 9 років тому +441

    I swear! This man saved me from going insane and failing my Algebra 2 class during the summer! The teacher was so awful. Thank you!

  • @aryashivakumar
    @aryashivakumar 2 роки тому +35

    at 4:12 you said "you get negative five" but the numbers are 2 and -3, so 2-3-11+6=-6, still not zero, but just clarifying in case anyone else caught that

    • @troydelantar7899
      @troydelantar7899 2 роки тому +2

      I have also noticed that, because 2 - 3 is equal to -1.

    • @UnbreakingSOUL
      @UnbreakingSOUL Рік тому +1

      I NOTICED THAT OMG I JUST SAW THIS COMMENT, weird is that I also put 4:12 👽 very timing

    • @UnbreakingSOUL
      @UnbreakingSOUL Рік тому

      @@troydelantar7899 maybe its just a mis-understanding

    • @momo-debest
      @momo-debest 4 місяці тому +1

      he also made a mistake at 5:12

  • @jvaniee
    @jvaniee 11 років тому +20

    Use the rational roots test and then plug the possible roots back in the polynomial to see if they = 0. Or, like what your professor did use the possible roots and the polynomial in synthetic division. If the remainder is 0 then it's a root. Both ways work but it'll probably be faster to do synthetic division.

  • @assassinblackop
    @assassinblackop 10 років тому +326

    You're doing God's work.

  • @nespecialistin7676
    @nespecialistin7676 10 років тому +4

    Man, you explain math better than all of the math teachers that I used here in Bulgaria. Thanks bro!

  • @Mahmzo
    @Mahmzo 5 років тому +41

    It seriously annoys me how I learn concepts in under 3 mins on this channel 😂😂

  • @thelasagneman6184
    @thelasagneman6184 6 років тому +2

    goes through 12 years of math, never hears of this, hears it, is ever so grateful, you sir are a legend!

  • @andreadelossantos164
    @andreadelossantos164 8 років тому +63

    I think it's not -27-33+6 it should be 27-33+6. Just get rid of the negative sign in 27 .

    • @promastrouvant3002
      @promastrouvant3002 7 років тому

      Andrea Delos Santos thats -27+-33

    • @charlieji8781
      @charlieji8781 6 років тому

      u r rite

    • @patrickdeveloper7186
      @patrickdeveloper7186 4 роки тому +2

      @promas this is misleading information and people will get confused, please correct it. Or further explain it.

    • @zorenmaniego5698
      @zorenmaniego5698 4 роки тому +2

      @@patrickdeveloper7186 at 5:30, The guy subtract 54 and 27 in the '54-27-33+6'. 54-27 is positive 27, not negative 27. That's why the 27 in his -27-33+6 should be positive. But since he did calculated the 27 as positive despite it visually doesn't, he answered the equation correctly.

    • @slysal5
      @slysal5 3 роки тому +2

      54-27= -27 ?????????????

  • @aadidesai1340
    @aadidesai1340 5 років тому +10

    5:31
    54-27=-27
    -27-33=-6
    -6+6=0
    Nice for you to help us, but please recheck work. + for the effort

  • @stormzsamurai
    @stormzsamurai 4 роки тому +4

    This video is helpin saving my grade in alg 2 rn. Thank you so damn much man. I have no idea when this came out but im so happy this is here!!

  • @sirruffalot
    @sirruffalot 9 років тому +2

    Patrick...I love you.. I've been watching your videos for quite some time now, you helped me through algebra, stats, precalc, calc etc..

  • @projux5778
    @projux5778 8 років тому +43

    You sir, are the reason I will pass my math quiz tomorrow.

  • @tanggapandaniela8342
    @tanggapandaniela8342 9 років тому +103

    Correct me if Im wrong, isn't it 2-3-11+6=-6?? Not 10? From the first equation u gave.

    • @jelly6023
      @jelly6023 8 років тому +8

      I was thinking that only I noticed it, made me do a double-take

    • @charlieliu7719
      @charlieliu7719 8 років тому +1

      Same

    • @nInOwarrior0312
      @nInOwarrior0312 8 років тому +1

      Unless one of them is suppose to be positive/negative, also remember PEMDAS

    • @Maryz0lSunny
      @Maryz0lSunny 6 років тому +1

      Yeah it's -6

    • @gabriel922987
      @gabriel922987 5 років тому +1

      U right fam

  • @NeatherBen221
    @NeatherBen221 5 років тому +1

    This guy saved me from failing my Algebra 2 class during this year! The way the Connexus program is so sketchy! Thank you so much!!!

  • @patrickjmt
    @patrickjmt  15 років тому +5

    if you are using real numbers, one can not just look at the degree and conclude the polynomial has that many roots.
    consider f(x) = x^2 + 4
    this has no roots (again, using real numbers).
    the degree is the maximum possible number of real roots.
    euler (i think) proved that there will be n solutions (some repeated) to a n degree polynomial if we use complex numbers - it was his phd thesis!

  • @vetacoth
    @vetacoth 12 років тому +1

    You are the best math teacher on UA-cam. Without you, I would have failed so many classes but I have to ask, when will this ever be applied to life?

  • @samsteve9220
    @samsteve9220 10 років тому +9

    Exactly what I was looking for

  • @miker3701
    @miker3701 11 місяців тому

    Studying day before major assessment. This is the best vid I found after an hour of looking. Thanks man

  • @thinkfree5008
    @thinkfree5008 7 років тому +1

    I have my test in less than 1hr and from the bottom of my heart thank you so much.

  • @felxsvoice1848
    @felxsvoice1848 7 місяців тому

    Just found your videos thru my college algebra class. You're so much more help than my teacher herself...

    • @patrickjmt
      @patrickjmt  6 місяців тому

      glad i could help you :)

  • @Jaxamuss
    @Jaxamuss 8 років тому +19

    I'm sure it's not to important now but he should have used synthetic division to check. it's a lot faster. if you don't know what synthetic division is, look it up, it's very helpful for dividing polynomials

    • @aneesak-ar-im3869
      @aneesak-ar-im3869 5 років тому +3

      JaxamussAwesomessMaximuss of course he could’ve used it but this is the rational root theorem that he’s trying to teach us (Yh I know this is like 2 years late oops)

    • @Supunay
      @Supunay 5 років тому

      @@aneesak-ar-im3869 no such thing as late on youtube bb

    • @CrimsonKnightmare1
      @CrimsonKnightmare1 5 років тому +3

      Ive heard synthetic divion is very limited

    • @mikuculus3720
      @mikuculus3720 4 роки тому

      CrimsonKnightmare How so?

  • @ShrimpWizard
    @ShrimpWizard 6 років тому +2

    You saved my Math grade ,Sir!! Thank you so much for such an awesome video!!

  • @konopong
    @konopong 15 років тому +6

    Just to mention, after you have determined the three roots, when you write f(x) you need to include a multiplicative constant factor, which turns out to equal 2. That will give you the function you started with.
    Another way to find the roots and factored form of f(x) is that once you have found one root you can divide (syn. div.) by that factor to determine the remaining quadratic, and then factor the quadratic.

    • @anamikajha7210
      @anamikajha7210 2 роки тому +1

      Drop a confident reply if you got me today after so many years ❤

  • @rithvik
    @rithvik 12 років тому

    Not trying to say Patrick's teaching skills are any less, but part of the supplement to his teach skills is the ability to rewatch and pause his videos to learn at your own pace. This is an added bonus to his skill to easily explain math without using words that are superfluous, as most math theorems do. Thanks so much for helping me out, been watching since 7th grade Alg 1, now am a freshman in Precalc :)

  • @DraoJasra
    @DraoJasra 9 років тому +2

    Thank you, so much. Your videos help me with math more than any teacher I've ever had

  • @patrickjmt
    @patrickjmt  14 років тому +3

    glad to help!

  • @Coughi3
    @Coughi3 3 роки тому +1

    I think at 6:41 , if you expand the bottom function, it doesn't equal the top function. Instead it equals the top function / 2. If you replace (x - 1/2) with (2x - 1) then it works. :)
    .
    .
    .
    .
    .
    ya

  • @Adriennee99
    @Adriennee99 9 років тому +3

    Thank you it made things a lot clearer and connected everything I learned so far

  • @chloecox7715
    @chloecox7715 4 роки тому +2

    Your videos are a blessing for me during this covid 19 crisis. THANK YOU!!!

  • @よりいち-v4g
    @よりいち-v4g Рік тому

    I was near 3 months old when this vid was made now it’s helping me after 14 years

  • @patrickjmt
    @patrickjmt  16 років тому +2

    yep! i hope it made sense for you : )

  • @meredithcamper3192
    @meredithcamper3192 4 роки тому +1

    At time -1.23 you note that 57-27=-27 and then go on the day that -27-33=-6, just thought I would note that 57-27 is in fact positive 27. Thank for making this video though!

  • @athenstube
    @athenstube 11 років тому +2

    You are really helping me along the path to my math degree. Thanks Patrick!

  • @Kapil65984
    @Kapil65984 Рік тому +1

    Superb i am watching it after 11 year

  • @ClaireDianneRamos-rw2wu
    @ClaireDianneRamos-rw2wu Рік тому

    I was exactly 4 months old when this video was made. And now I'm 15 years old and this video helps me.

  • @mehran528
    @mehran528 Рік тому +1

    I was 1 year old when this video was made and now 14years later this video is helping me😀😀

  • @spinel-lipatashaneloreala.6200
    @spinel-lipatashaneloreala.6200 3 роки тому

    i wasnt ever bornn when this video was made.. then it helped me 13years later thank you

  • @cryomancer20
    @cryomancer20 4 роки тому +1

    I was overthinking this cause the other videos made it feel complicated. Thanks for explaining it straight on :)

  • @manyasuhag67
    @manyasuhag67 2 роки тому

    for some reason my math class books and other material teaches weirdly and complicates stuff. This was very helpful!

  • @negosyante123
    @negosyante123 3 роки тому +2

    Be careful when you are adding or subtracting negative signs.
    It's actually not -27 but +27-33+6
    =0

    • @StormForthcoming
      @StormForthcoming 2 роки тому +1

      ohhhh thats why i was confused lol

    • @negosyante123
      @negosyante123 2 роки тому

      @@StormForthcoming yeah, you can visit my profile as well but I don't think so if you can understand because I used Filipino language.

  • @erinrisinger2413
    @erinrisinger2413 9 років тому

    Why are you not my teacher, you make it so simple and easy to understand.

  • @patrickjmt
    @patrickjmt  2 місяці тому +2

    Hi all! Please post comments, questions and anything else on your mind in the comment section! Also, don’t forget to LIKE, THUMBS UP, and SUBSCRIBE! I’d appreciate it greatly :)

  • @yourvideosaregomi899
    @yourvideosaregomi899 10 років тому +128

    U put negative 27 even though it's positive

    • @khomchat1
      @khomchat1 6 років тому

      Nope - 3(3)`2 is 27 since 3*3=9 and 9*-3=-27

    • @EstiagoGaming
      @EstiagoGaming 6 років тому +11

      @@khomchat1 he is taling about 54-27 =27 not -27

    • @khomchat1
      @khomchat1 6 років тому +2

      Oops looking at the wrong one

    • @edmundsin4862
      @edmundsin4862 5 років тому +10

      Yes, he wrong -27 but he still computed it as positive 27. So no actual error that affected the answer.

    • @mr.lemonjuice9098
      @mr.lemonjuice9098 5 років тому +1

      But the answer still is correct

  • @B.W.L.9786
    @B.W.L.9786 4 роки тому

    Video is 11 years old but still better then my math teacher thanks man!

  • @tyobrien1340
    @tyobrien1340 2 роки тому +2

    You just explained in 6 minutes what my teacher couldn't explain in 30. Thank you.

  • @unfaithful1034
    @unfaithful1034 10 років тому +14

    You explained a lot better than my package of notes.
    Thanks!

  • @fmalove2
    @fmalove2 12 років тому +1

    Thank you so much! I had a hard time with figuring out where the fractions came from, but now I know!

  • @WLOfails
    @WLOfails 13 років тому

    Thank you so much! I never knew how simple this type of problem could be and now I can start my homework... before school reopens tommorow...

  • @patrickjmt
    @patrickjmt  15 років тому +2

    yes, first factor out the x.
    then you can use rational roots thm on the polynomial of degree 6

    • @haiderkazmi8806
      @haiderkazmi8806 Рік тому

      i was 12 years old when you uploaded this video and now i am 26

  • @davidhuang5645
    @davidhuang5645 10 років тому

    Best math teacher on UA-cam.

  • @dsps301
    @dsps301 11 років тому +2

    I wanna take a minute to thank you, for spending time to help me, and everyone else who has seen your videos! Keep making them! :D

  • @UtuberKARAM
    @UtuberKARAM 15 років тому +1

    OMG you made my life so much easier i have a test in 3 hours... i was gna FAIL thank you thank you thank you keep up the great work :D

  • @cynthiajones5116
    @cynthiajones5116 4 роки тому

    This video helped me explain this concept to my high school students.

  • @patrickjmt
    @patrickjmt  13 років тому +3

    @Zen113 what i am doing on youtube is more effective than what i could do in any class room i think.

  • @xXwannabeguitaristXx
    @xXwannabeguitaristXx 14 років тому

    I wish you were my math teacher.
    I mean, I understand everything I needed to understand by watching your short videos than being in a room for an hour and a half!
    :)

  • @sovietwizard1620
    @sovietwizard1620 8 місяців тому

    Theres a faster way to do this which is if it's a cubic (90% of the time it will be a cubic, since those appear more)
    1. Simplify as much as possible, make sure there are only integers. MAKE SURE "a" coefficient is positive (Multiply by -1 if not)
    2. Try 1 by adding all coeeficients, and try -1 (You only need one root to end this cycle of rational root theorem).
    3. If no answers, THEN Take descartes's rule of signs since this works for cubics as it yields 2 possible sets only.
    *NOTE: A cubic is depressed if there is no x^2 part.
    4a. IF the cubic is NOT depressed Take the derivative and set to 0, then plug the answer back into the equation using a calculator and see which of the de moivre's part is right. If its complex answer there can max be 1 root. If one of the answers yields 0 as y, it means that the equation has a double-root and that is one of the roots. You can synthetic divide and stop here if this is true. If it yields 0 but there is only 1 horizontal tangent, it means there is a triple-root and you have just found all 3 roots (this is super duper rare though)
    4b. IF the cubic IS depressed, you can choose to either do 4a or find the discriminant and use that. If the discriminant is above 0 there is 1 real root, below 0 and 3 real roots, equal to 0 and 2 real roots. (Search up how to find the discriminant)
    5. Using that info determine which of Descartes possibilities are right.
    6. Use #5 and synthetic division and figure out if its below or above zero. If its below zero, eliminate all of the answers below it (unless there is another root, then be careful) and same for if it's above. (If theres another root, the graph can go back the other way again, so it's important to know it). Using this will eliminate like 2-3 solutions every try.
    7. IF there is a rational root, find the leftover quadratic and figure out the other roots. (If its 1 real root, there will be 2 complex roots, and if its 3/2 roots there will be 2 real roots.)
    8. If there is no rational root, either 1. Die 2. Depress the cubic and use Cardano's formula and die of exhaustion 3. Learn to code and approximate the root. (don't be that guy)
    Hopefully this helps, if it's not a cubic then die as well!

  • @patrickjmt
    @patrickjmt  16 років тому +10

    i have a video on synthetic division if that helps : )

    • @manshagoel
      @manshagoel 3 роки тому

      Do you have one for integer root test/theorem

  • @eberever3930
    @eberever3930 7 років тому

    god bless your soul mister. you have literally just saved me from failing my math class😭😭😭👌👏

  • @Neighthann
    @Neighthann 12 років тому

    bro you rock i learn more math on you tube than in class somtimes

  • @herniethetwig7994
    @herniethetwig7994 5 років тому

    Thanks for teaching me what my teacher didn’t

  • @liyan2168
    @liyan2168 5 років тому

    My Maths teacher sucks at an epic level... Thank u!!! I understood now

  • @MarsTheProgrammer
    @MarsTheProgrammer 12 років тому

    thank you so much sir, without you, i wouldn't pass calculus and therefore not be able to get into computer systems classes in college.

  • @ШерозАлижанов
    @ШерозАлижанов 4 роки тому

    dude, thank you very much, next month I will have an entrance exam in mathematics at the Lyceum, and this video helped me understand the topic, thanks)))

  • @WestSTL
    @WestSTL 15 років тому +1

    its almost 1 a.m. and I have a quiz tomorrow in honors alg 2 on this and I had no idea what was going on until this video. THANK YOU! you accomplished in 6 mins what my douche teacher couldn't in 90 minutes

  • @patrickjmt
    @patrickjmt  15 років тому +1

    no, u can always just reorder it

  • @Oaishe
    @Oaishe 11 років тому +2

    Thank you soo much for this video!! I was so lost but now I understand! Thank you!

  • @strugglingcollegestudent
    @strugglingcollegestudent Рік тому

    I am a linear algebra student who is not allowed a calculator so I am back to the basics for finding the characteristic equation of a 3x3 (not triangular) matrix

  • @brucezhou4655
    @brucezhou4655 8 років тому +11

    the last polynomial should be multiplied by 2, which is 2(x-1/2)(x+2)(x-3)
    anyway thanks a lot

  • @jackjik5076
    @jackjik5076 11 років тому +1

    You helped me get an A in this chapter, thanks.

  • @backpacker_mac
    @backpacker_mac 11 років тому

    Awesome! I am sick and I am still learning the stuff my classmates are in class. Awesome!!

  • @wayofwonderer5929
    @wayofwonderer5929 Рік тому

    All your videos Such a valuable gift 🎁 for me on youtube .

  • @drewskiidrew6258
    @drewskiidrew6258 4 роки тому +9

    What am I in school for when you exist?

  • @junqianchen7128
    @junqianchen7128 11 років тому

    you are much better than my math teacher!! thanks a lots!!

  • @ryanholmes5294
    @ryanholmes5294 11 років тому

    You are a life saver for my algebra 2 final!

  • @maddygolier3210
    @maddygolier3210 3 роки тому

    12 years later and still killing the game

  • @XgtkidX
    @XgtkidX 14 років тому +1

    thanks so much. helped me tons with my finals!

  • @azmoto-in5116
    @azmoto-in5116 7 років тому +1

    Thank you very much you are a very pro efficient teacher!

  • @yougotnojams5182
    @yougotnojams5182 4 роки тому

    thank you so much,sir! You're much better than my math book !

  • @raikantopeni9532
    @raikantopeni9532 4 роки тому

    You made this very easy. Thanks

  • @NametagstheNametag
    @NametagstheNametag 12 років тому

    If you become an accountant, when will you ever apply chemistry or biology in life? If you become a professional football player, when will you ever apply world history in your line of work? Pretty much everything you learn in school is built to open up more paths for your future and what you will use depends on what you become.

  • @夜雨声醒
    @夜雨声醒 6 років тому +2

    My teacher really confuses me, now I understand thank you.

  • @DaDerpyCarrot
    @DaDerpyCarrot 4 роки тому

    thank you, my confusion is relieved

  • @charlieliu7719
    @charlieliu7719 8 років тому

    Ur 1 of the 2 peoples vids I watch for math, the other is David reneau, but u still do really do it good, since David teaches a class, he makes more clear for me tho

  • @イエル-x9w
    @イエル-x9w 2 роки тому

    thanks so much for the solution :) I was having a hard time using synthetic division. I'll try doing how he's doing.

  • @_A_343
    @_A_343 7 років тому +1

    This video helped a great deal. Thanks you.

  • @oscarmeadowcroft
    @oscarmeadowcroft 3 роки тому

    The concept of the rational roots testing makes sense, but factorising that polynomial gives (2x-1)(x+2)(x-3)

  • @patrickjmt
    @patrickjmt  16 років тому +2

    yes, i just need to remake this one later today! : )

  • @predatorzealot1000
    @predatorzealot1000 9 років тому

    didnt know it was uploaded in 2008 but thank you so much for uploading this video
    it really helped a lot

  • @gabbyjay8581
    @gabbyjay8581 5 років тому

    this is a really lengthy process, but it’s the only thing i understand so thank you :))

  • @theecreativesoul
    @theecreativesoul 15 років тому

    Thanks! My teacher taught this in a terrible way. You are so much help.

  • @centrax4716
    @centrax4716 4 роки тому

    thanks man you explained it really well

  • @francisr8205
    @francisr8205 8 років тому +3

    Shouldnt you consider the negative versions of the possible rational roots too?

  • @KillerTacos54
    @KillerTacos54 4 роки тому

    What a coincidence, our maths holiday homework had this exact question haha

  • @iankelley9302
    @iankelley9302 3 роки тому +2

    I'm in algebra three and we never were taught this.

  • @julianalinares9825
    @julianalinares9825 10 років тому

    thank youuu ! you simplify things so muchh

  • @RockPaperScissor267
    @RockPaperScissor267 3 роки тому

    Thanks, easy to understand.

  • @janellehafalla3659
    @janellehafalla3659 8 років тому +1

    this literally saved my life

  • @brennan-qc5qt
    @brennan-qc5qt 4 роки тому

    I got a 40% on my test. Thank you

  • @RidinMusicLines
    @RidinMusicLines 13 років тому

    @aceonx the amount of roots depends on the highest power x factor... so in x^3 + x^2 + x + 4 or something, 3 is the amount of roots, because this is the highest power x factor