Finding the Remainder in Division of Polynomials | Step-by-Step Solution

  • Thread starter Ahmed Abdullah
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In summary, the conversation discusses finding the remainder when a polynomial is divided by another polynomial. The attempt at a solution involves using a quotient and remainder, but the person is unsure of how to proceed. The concept of defining a norm on polynomials is also mentioned. Someone suggests solving the problem by hand.
  • #1
Ahmed Abdullah
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What is the remainder??

Homework Statement



What is the remainder when (a+b+c)^333-a^333-b^333-c^333 is divided by (a+b+c)^3-a^3-b^3-c^3?

Homework Equations



None

The Attempt at a Solution



I tried this
(a+b+c)^333-a^333-b^333-c^333 = Q{(a+b+c)^3-a^3-b^3-c^3}+h

where h is the remainder
I proceeded further but only managed to work out that Q>(a+b+c)^330.
I don't know how to attack this types of problem.
Please help!
 
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  • #2
Are these polynomials in the variables a,b,c? If so, I don't think there's a natural way to talk about remainders. You first have to define a norm on the set of polynomials. For example, you could take it to be the largest power of a (or b, or c, which is what I mean by when I say there's no natural way).
 
  • #3
Why Don't you just do it by hand? :p
 

Related to Finding the Remainder in Division of Polynomials | Step-by-Step Solution

What is the definition of "Finding the Remainder in Division of Polynomials"?

Finding the remainder in division of polynomials is the process of dividing two polynomials and determining the remainder of the division. The remainder is the polynomial that is left over after dividing the larger polynomial by the smaller polynomial.

What is the purpose of finding the remainder in division of polynomials?

The purpose of finding the remainder in division of polynomials is to simplify the original polynomial division problem. By finding the remainder, we can express the original polynomial as a quotient plus a remainder, which is often a simpler and more manageable form.

What are the steps to finding the remainder in division of polynomials?

The steps to finding the remainder in division of polynomials are as follows:

  1. Arrange the polynomials in descending order.
  2. Determine the highest degree term in the divisor and divide it into the highest degree term in the dividend.
  3. Multiply the quotient by the divisor.
  4. Subtract the result from the dividend.
  5. Repeat steps 2-4 until all terms in the dividend have been divided.
  6. The final result is the remainder.

Can you provide an example of finding the remainder in division of polynomials?

Yes, for example, if we have the problem (x^3 + 2x^2 + 3x + 4) ÷ (x + 1), we would begin by dividing the highest degree term (x^3) by the highest degree term in the divisor (x), resulting in a quotient of x^2. We then multiply x^2 by (x + 1), giving us (x^3 + x^2). We subtract this from the original dividend to get (x^2 + 3x + 4). We then repeat the process with the next highest degree term (x^2), resulting in a quotient of x and a remainder of (2x + 4). Our final result is (x^2 + x + 2) with a remainder of (2x + 4).

Are there any common mistakes to avoid when finding the remainder in division of polynomials?

Yes, common mistakes to avoid when finding the remainder in division of polynomials include:

  • Forgetting to arrange the polynomials in descending order.
  • Dividing the wrong terms or dividing by a term that does not divide evenly into the dividend.
  • Forgetting to multiply the quotient by the divisor before subtracting.
  • Not repeating the process for all terms in the dividend.
  • Forgetting to include the remainder in the final result.

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