What is a good technique for solving polynomials?Say for example: 6 x 3 − 17...

capellitad9

capellitad9

Answered

2022-07-16

What is a good technique for solving polynomials?
Say for example:
6 x 3 17 x 2 4 x + 3 = 0

Answer & Explanation

tiltat9h

tiltat9h

Expert

2022-07-17Added 14 answers

First, to minimize the lead coefficient, we reverse the polynomial (which reciprocates the roots) and then we scale it to make the leading coefficient 1, i.e. we apply the AC-method, which yields
f ( x ) = 3 x 3 4 x 2 17 x + 6     9 f ( x ) = ( 3 x ) 3 4 ( 3 x ) 2 51 ( 3 x ) + 54 =     z 3 4 z 2 51 z + 54 ,       z = 3 x
Note z = 1 is a root. By Vieta, other roots have product 54 / 1 , sum 4 1 = 3 , so are 9 , 6.
Hence z = 1 , 9 , 6 so x = z / 3 = 1 / 3 , 3 , 2 , which reciprocated yields 3 , 1 / 3 , 1 / 2.

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