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UnkleRhaukus

  • 2 years ago

roots of a trinomial \[ax^3+bx^2+cx+d=0\] let the roots be \(p,q,r\) When \(x\) is equal a root the equation equals zero \[(x-p)(x-q)(x-r)=0\] \[(x-p)(x^2-(q+r)x+qr)=0\] \[x\left(x^2-(q+r)x+qr\right)-p\left(x^2-(q+r)x+qr\right)=0\] \[\left(x^3-(q+r)x^2+(qr)x\right)-\left(px^2-p(q+r)x-qpr\right)=0\] \[x^3-(p+q+r)x^2+(qr+pq+pr)x-pqr=0\] comparing with \[x^3+\frac bax^2+\frac cax+\frac da=0\] \[-(p+q+r)=\frac ba\qquad\text{the sum of the roots is } -\frac ba\] \[(qr+pq+pr)=\frac ca\qquad\text{sum of products of pairs of the roots is } \frac ca\] \[-pqr=\frac da\qquad \text{the product of the roots is} -\frac da \]

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  1. mahmit2012
    • 2 years ago
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    |dw:1351834318182:dw|

  2. mahmit2012
    • 2 years ago
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    |dw:1351834441300:dw|

  3. mahmit2012
    • 2 years ago
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    I divided to a.

  4. UnkleRhaukus
    • 2 years ago
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    what about complex solutions

  5. mahmit2012
    • 2 years ago
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    |dw:1351834556416:dw|

  6. UnkleRhaukus
    • 2 years ago
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    i guess a trinomial always has at least one real root

  7. mahmit2012
    • 2 years ago
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  8. mahmit2012
    • 2 years ago
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    |dw:1351834848416:dw|

  9. hartnn
    • 2 years ago
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    if the trinomial has real co-efficients, then complex roots occur in pairs, hence it must have atleast 1 real solution .

  10. mahmit2012
    • 2 years ago
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  11. UnkleRhaukus
    • 2 years ago
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    i see

  12. mahmit2012
    • 2 years ago
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    |dw:1351835035013:dw|

  13. mahmit2012
    • 2 years ago
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    |dw:1351835192802:dw|

  14. mahmit2012
    • 2 years ago
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    |dw:1351835381933:dw|

  15. mahmit2012
    • 2 years ago
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    this is the general for all n-order polynomial equation.

  16. UnkleRhaukus
    • 2 years ago
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    awesome

  17. hartnn
    • 2 years ago
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    notice the sign change everytime, product of all roots = \( (-1)^n a_0/a_n \)

  18. mahmit2012
    • 2 years ago
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  19. mahmit2012
    • 2 years ago
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    I should mention that there are all real and complex roots in formulas.

  20. mahmit2012
    • 2 years ago
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    |dw:1351839125496:dw|

  21. mahmit2012
    • 2 years ago
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    |dw:1351839193349:dw|

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