UnkleRhaukus
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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mahmit2012
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mahmit2012
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mahmit2012
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I divided to a.
UnkleRhaukus
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what about complex solutions
mahmit2012
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UnkleRhaukus
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i guess a trinomial always has at least one real root
mahmit2012
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mahmit2012
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hartnn
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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 .
mahmit2012
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UnkleRhaukus
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i see
mahmit2012
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mahmit2012
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mahmit2012
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mahmit2012
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this is the general for all n-order polynomial equation.
UnkleRhaukus
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awesome
hartnn
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notice the sign change everytime,
product of all roots = \( (-1)^n a_0/a_n \)
mahmit2012
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mahmit2012
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I should mention that there are all real and complex roots in formulas.
mahmit2012
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mahmit2012
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