A previous question asked about a root of:
f(x) = 4x^7 +6x^6 +3x^5 +250x^4 +1000x^3 +75x^2 -20x +17
Can you estimate or generalize approximately about the roots, using only, pencil and paper, factorizations or approximations? (no calculators, computers, solvers or Newton-Raphson. See how far you can get using your head!).
Can you at least bound the regions in which each root lies, by identifying the zero-crossings of f?
Two ideas: a) since all terms other than "-20x" are positive, most roots will be negative.
Hence it might make it algebraically easier to substitute u→(-x) and consider
f(u) = -4u^7 +6u^6 -3u^5 +250u^4 -1000u^3 +75u^2 +20u +17
b) I'm thinking one promising way might be to regard it as the superposition of two (or more) polynomials, a high-order one and a lower-order one.
Your thoughts?
[ http://answers.yahoo.com/question/index?qid=20071215210154AAY3BhC ]
2007-12-15
17:15:55
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2 answers
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asked by
smci
7