Applying the Remainder Theorem
Date: 11/19/2005 at 18:13:33 From: Craig Subject: Remainder Theorem Given f(x) = 7x^4 + 9x^3 + 4x^2 - 4x + 16, use the Remainder Theoreom to find f(2). I don't understand the theorem as a whole.
Date: 11/20/2005 at 00:09:37 From: Doctor Wilko Subject: Re: Remainder Theorem Hi Craig, Thanks for writing to Dr. Math! The remainder theorem says that when a polynomial f(x) is divided by (x-r), the remainder is f(r). In other words, the remainder of the division process has the same value you would get by plugging r into the function and evaluating it. So the remainder theorem, in your case, says that when the polynomial f(x) = 7x^4 + 9x^3 + 4x^2 - 4x + 16 is divided by (x - 2), the remainder is the same as what you get by evaluating f(2). Let's see how to use this. Your question is something like, "Use the remainder theorem to evaluate f(2)." You're supposed to do the following: Use division (synthetic or long) to divide the polynomial f(x) by (x-2) and note the remainder of 208. I recommend synthetic: 2 | 7 9 4 -4 16 14 46 100 192 -------------------- 7 23 50 96 208 From this (because of the theorem), you can state that f(2) = 208 (without actually calculating f(2) directly). Why is this useful? f(2) is somewhat complicated to calculate by hand with all the exponents, but I can get the answer to f(2) by doing a simpler problem (division of the polynomial f(x) by (x-2)) and taking note of the remainder. This remainder is the answer to f(2). The remainder theorem gives me an alternate way to calculate f(2). -------------------------------- This also leads into another theorem, the factor theorem. Basically the factor theorem says that if you take a polynomial f(x) and calculate f(r), and f(r) = 0, then (x-r) is a factor of f(x). To relate this to your question, is f(2) = 0? No. We found that f(2) = 208. Therefore (x-2) is not a factor of f(x) = 7x^4 + 9x^3 + 4x^2 - 4x + 16. This means that (x-2) doesn't divide evenly into the polynomial f(x). To summarize this, the factor theorem gives you a quick way to see if some (x-r) is a factor of a polynomial f(x). Just calculate f(r). If f(r) = 0, then (x-r) is a factor of f(x). --------------------------------- If you look back at these two theorems, you'll see that they are closely related. Here's a couple more links on these topics from our archives: Applying the Remainder Theorem http://mathforum.org/library/drmath/view/53203.html Factoring and the Factor Theorem http://mathforum.org/library/drmath/view/53146.html Does this help? Please write back if you have further questions. :-) - Doctor Wilko, The Math Forum http://mathforum.org/dr.math/
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