Factor theorem

For p,q,r,s,a,b,c,d in Z-{0}, suppose px³ + qx² + rx + s = 0.

 

If px³ + qx² + rx + s  can transform (ax - b)(cx - d)(ex - f),

we get

             px³ + qx² + rx + s = (ax - b)(cx - d)(ex - f)

                                           = acex³ +   …   -bdf

 

Compared with both sides, p = ace, s = -bdf.

Therefore,

             s/p = -bdf / ace,

 

while the solutions of (ax - b)(cx - d)(ex - f) = 0 are

               x = b/a, d/c, f/e.

 

Thus  the candidates of solutions of the equation lies in s/p = -bdf / ace.

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