Method of detached coefficients division

In summary, the conversation discusses the process of dividing a polynomial by another polynomial using the method of detached coefficients. The correct placement of coefficients and variables is important in the quotient, which can be achieved through a process of aligning and lining up the coefficients with the powers of the variables. This method is also known as synthetic division and can be used for division by linear binomials with a leading coefficient of 1.
  • #1
drooble122
6
1
Homework Statement
Divide 15+2m^4-31m+9m^2+4m^3+m^5 by 3-2m-m^2
Relevant Equations
Use method of detached coefficients
-1 -2 3 | 1 2 4 9 -31 15 | -1 -7 5
1 2 -3
7 9 -31 15
7 14 -21
-5 -10 15
5 -10 15The quotient has the coefficients -1, -7, 5. Now here is the problem. I assumed the variables would thus be -1m^2-7m-5. Turns out the correct answer is -1m^3-7m-5. But i thought in the quotient its all lined up from right to left by the power: 0,1,2,3...etc. Clearly this is not the case.
Any instruction on how to assign the proper power to the variable would be appreciated. Thanks.
 
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  • #2
drooble122 said:
Homework Statement:: Divide 15+2m^4-31m+9m^2+4m^3+m^5 by 3-2m-m^2
Relevant Equations:: Use method of detached coefficients

-1 -2 3 | 1 2 4 9 -31 15 | -1 -7 5
1 2 -3
7 9 -31 15
7 14 -21
-5 -10 15
5 -10 15The quotient has the coefficients -1, -7, 5. Now here is the problem. I assumed the variables would thus be -1m^2-7m-5. Turns out the correct answer is -1m^3-7m-5. But i thought in the quotient its all lined up from right to left by the power: 0,1,2,3...etc. Clearly this is not the case.
Any instruction on how to assign the proper power to the variable would be appreciated. Thanks.
You need zero as a place holder.
Line stuff up with "code".

Code:
   -1 -2 3 |       1 2  4  9  -31 15 | -1 0 -7 5
                   1 2 -3
                  ______
                     0  7   9  -31 15
                     0  0   0
                    ________
                         7   9  -31 15
                         7  14  -21
                         _________
                             -5  -10 15
                             -5  -10 15
 
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  • #3
SammyS said:
You need zero as a place holder.
Line stuff up with "code".

Code:
   -1 -2 3 |       1 2  4  9  -31 15 | -1 0 -7 5
                   1 2 -3
                  ______
                     0  7   9  -31 15
                     0  0   0
                    ________
                         7   9  -31 15
                         7  14  -21
                         _________
                             -5  -10 15
                             -5  -10 15
Thanks for this. Here I think this process is called 'synthetic division'. And I never understood it. Until now!
 
  • #4
neilparker62 said:
Thanks for this. Here I think this process is called 'synthetic division'. And I never understood it. Until now!
This isn't what I know as synthetic division. This is long division, done with a little bit of "short-hand", keeping track coefficients in columns to eliminate the need for repeatedly writing the variable to various powers.

To my knowledge, synthetic division is restricted to division by linear binomials with leading coefficient of 1 .
 
  • #5
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Likes SammyS

Related to Method of detached coefficients division

1. What is the Method of Detached Coefficients Division?

The Method of Detached Coefficients Division is a mathematical technique used to divide polynomials by another polynomial. It involves breaking down the dividend into smaller parts and dividing each part separately by the divisor.

2. How is this method different from traditional long division?

The Method of Detached Coefficients Division is different from traditional long division because it involves breaking down the dividend into smaller parts and dividing each part by the divisor, rather than dividing the entire dividend at once. This method can be more efficient and easier to understand for more complex polynomials.

3. What are the steps involved in this method?

The steps involved in the Method of Detached Coefficients Division are as follows:

  1. Arrange the dividend and divisor in descending order of their exponents.
  2. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
  3. Multiply the first term of the quotient by the entire divisor and subtract it from the first part of the dividend.
  4. Repeat the process for the remaining terms of the dividend, adding the results to the quotient.
  5. If there is a remainder, bring it down and continue the process until the remainder is zero or of a lower degree than the divisor.

4. What are the advantages of using this method?

The Method of Detached Coefficients Division has several advantages:

  • It can be more efficient and less time-consuming than traditional long division, especially for more complex polynomials.
  • It can be easier to understand and follow for those who struggle with traditional long division.
  • It can be used to divide polynomials with any degree, whereas traditional long division can only be used for polynomials with a degree of two or more.

5. Are there any limitations to this method?

While the Method of Detached Coefficients Division is a useful technique, it does have some limitations:

  • This method can only be used for dividing polynomials by another polynomial, not by a monomial or a number.
  • If the divisor has more than one term, it must be in standard form (descending order of exponents) for this method to work.
  • If the dividend has missing terms, it may be necessary to add placeholder terms with a coefficient of zero for this method to work.

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