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swissgirl

  • 2 years ago

Reduce the polynomial and find its complex roots \(f(x)=x^4+5x^3-9x^2-85x-136\)

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  1. swissgirl
    • 2 years ago
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    lol didnt know u were on

  2. swissgirl
    • 2 years ago
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    not quite sure how to reduce it since u start dealing with decimals. It shld be quite simple though

  3. KingGeorge
    • 2 years ago
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    By reduce do you mean factor?

  4. swissgirl
    • 2 years ago
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    i am assuming so

  5. KingGeorge
    • 2 years ago
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    Well, I have a couple ideas, so let's see if I can do this by hand... \[f(x)=x^4+5x^3-9x^2-85x-136\\ f(x)=x^4-17x^2+5x^3+8x^2-17(5x-8)\\ f(x)=x^4-17x^2+x^2(5x+8)-17(5x+8)\\ f(x)=x^4-17x^2+(x^2-17)(5x+8)\\ f(x)=x^2(x^2-17)+(x^2-17)(5x+8)\\ f(x)=(x^2-17)(x^2+5x+8) \]

  6. KingGeorge
    • 2 years ago
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    So you can use the quadratic formula on \(x^2+5x+8\) to find the complex roots, and just \[x^2-17=0\implies x=\pm\sqrt{17}\]For the real roots.

  7. swissgirl
    • 2 years ago
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    Let me just see if I can follow what u did

  8. KingGeorge
    • 2 years ago
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    Take your time. I'm honestly very surprised I was able to factor that almost entirely by hand. (I used wolfram to check correctness, and to find the gcd of 85, 136).

  9. swissgirl
    • 2 years ago
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    idk if its possible. Dont drive urself crazy buttt I cant decipher what is squred and what is tripled. Is it possible to make it larger

  10. swissgirl
    • 2 years ago
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    whoops cubed lol

  11. KingGeorge
    • 2 years ago
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    \[\Large f(x)=x^4+5x^3-9x^2-85x-136\\ \Large f(x)=x^4-17x^2+5x^3+8x^2-17(5x-8)\\ \Large f(x)=x^4-17x^2+x^2(5x+8)-17(5x+8)\\ \Large f(x)=x^4-17x^2+(x^2-17)(5x+8)\\ \Large f(x)=x^2(x^2-17)+(x^2-17)(5x+8)\\ \Large f(x)=(x^2-17)(x^2+5x+8)\]

  12. swissgirl
    • 2 years ago
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    Thanks :)

  13. KingGeorge
    • 2 years ago
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    welcome

  14. swissgirl
    • 2 years ago
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    OKKKK so like will this method always work?

  15. KingGeorge
    • 2 years ago
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    Not always. This is just factoring by grouping, and while potentially useful, it is often difficult to identify the correct time to use it, and what to group together. I just got lucky that I guessed the right things to group. For example, if I had \[\Large f(x)=x^4+5x^3-10x^2-85x-136\]instead, I would not be able to group it as easily.

  16. swissgirl
    • 2 years ago
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    ohhhh i seee. I guesss I have to play around. Not as simple as I wish it wld be. Thanks KG:)))))))))))

  17. KingGeorge
    • 2 years ago
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    You're welcome. I will note however, that if they tell you to factor something, it's usually possible to do it by hand with a little cleverness.

  18. swissgirl
    • 2 years ago
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    Haha So I hope and pray that it will be so. lol Thanks KGGGG

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