Remainder Theorem and Synthetic SubstitutionDate: 06/17/2004 at 19:58:44 From: Marc Subject: algebra and functions Let P(x) be the polynomial P(x) = x^15 - 2004x^14 + 2004x^13 - ... - 2004x^2 + 2004x Calculate P(2003). I really do not know where to start. Date: 06/18/2004 at 00:22:38 From: Doctor Greenie Subject: Re: algebra and functions Hi, Marc -- The Remainder Theorem tells us that the value of P(2003) is the remainder you get when you divide the given polynomial by (x - 2003). A quick exercise beginning that polynomial division process using synthetic division shows a pattern which makes it easy to find the answer to the problem without completing the synthetic division: 2003 | 1 -2004 +2004 -2004 +2004 -2004 +2004 ... | 2003 -2003 2003 -2003 2003 -2003 ... --------------------------------------------- 1 -1 1 -1 1 -1 1 ... The "remainder" is +1 after division of each term with an odd power and -1 after division of each term with an even power. Since the last term of the polynomial is an odd power, the value of P(2003) is +1. I hope this helps. Please write back if you have any further questions about any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ |
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