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 Subject: "Remainder Theorem" Previous Topic | Next Topic
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saideep
Member since Jun-1-02
Jun-01-02, 09:21 PM (EST)    "Remainder Theorem"

 There is a problem with the validity of the remainder theorem.The theorem states that if f(x) is divided by x-a, the remainder is f(a), where f is any rational integral function.But supposeing f(x) = x^2 + x + 5,f(4) will be 25.Supposing we divide f(4) with (4 - 3) i.e 1, the remainder should be f(3).But by actual division (25/1), the remainder is 0, but f(3) is 17. How is that possible. Surely, there must be something wrong.T SAIDEEPT SAIDEEP

amy guest
Jun-01-02, 11:37 PM (EST)

1. "RE: Remainder Theorem"
In response to message #0

 The remainder divided by the divisor is added to the quotient, so a remainder of 17 is divided by X-3. When X is no longer part of the equation the remainder is added to the quotient without being divided by a variable, so it can't be distinguished from the quotient. x^2+x+5/x-3 = x+4R17 = x+4+17/x-3in long division: x+ 4 + 17(R)/x-3x-3 /x^2 +x+ 5 -(x^2-3x) =4x+ 5 -(4x-12) =17with f(x)=4: 4+ 4+17(R)/4-3 = 4+4+17/1 = 4+4+17 = 254-3 /16+ 4+ 5 -(16-12) =16+ 5 -(16-12) =17hope this helps Bo Jacoby guest
Sep-13-02, 08:44 AM (EST)

2. "RE: Remainder Theorem"
In response to message #0

 >There is a problem with the validity of the remainder >theorem. >The theorem states that if f(x) is divided by x-a, the >remainder is f(a), where f is any rational integral >function. We use the letter-convention: Letter 'x' indicate a Variable.Letter 'a' indicate a Constant.Letter 'f' indicate a Function. So 'f(x)' means the function 'f' , not the number,while 'f(a)' means the function value in the point 'a'.'x-a' means a function of 'x', not a number.This convention is often used but rarely spoken of, because it cannot easily be avoided even if it'stinks.So, dividing f(x) by (x-a) means writing: f(x)=(x-a)g(x) + b . This is an equality of functions of x .f(x) is the dividend.x-a is the divisor.g(x) is the quotient. b is the remainder.Now substitute x=a to get f(a)=(a-a)g(a)+b = 0*g(a)+b = bSo, the remainder, b, of a function, f(x), divided by the function (x-a),equals the function value f(a).This was the theorem.>But supposeing f(x) = x^2 + x + 5, >f(4) will be 25. >Supposing we divide f(4) with (4 - 3) i.e 1, the remainder >should be f(3). No sir, you should divide the function f(x)with the function x-4 :f(x) = (x-4)(x+5)+25Check that (x-4)(x+5)+25 = x^2+x+5 .The dividend is x^2+x+5. The divisor is x-4.The quotient is x+5.The remainder is 25.The theorem then gives f(4)=25, which is correct.

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