Method of detached coefficients division

[drooble122]

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 15


The 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.

 

[SammyS]

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 15


The 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".

   -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

 

[neilparker62]

You need zero as a place holder.
Line stuff up with "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!

 

[SammyS]

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 .

 

[neilparker62]

Ok thanks for the correction. After some research I think I now understand synthetic division as a 'collapsed' version of the method of detached coefficients which you have demonstrated above.

https://courses.lumenlearning.com/i...use-synthetic-division-to-divide-polynomials/

They make a dog's breakfast of the polynomial division but it does show how detached coefficients 'evolve' into synthetic division.