Method of detached coefficients division

AI Thread Summary
The discussion centers on the method of detached coefficients for polynomial division, specifically addressing a misunderstanding about the assignment of variable powers in the quotient. The initial assumption was that the coefficients corresponded to powers of m in descending order, leading to confusion when the correct polynomial was identified as -1m^3 - 7m - 5 instead of -1m^2 - 7m - 5. Participants clarified that zero placeholders are necessary for proper alignment during the division process. There was also a distinction made between synthetic division and the method of detached coefficients, with synthetic division being a simplified version applicable only to linear binomials. Overall, the conversation provided insights into polynomial division techniques and the importance of correctly aligning coefficients.
drooble122
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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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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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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!
 
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 .
 
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