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QUESTION: Hello, I recently started a rigorous course in Math and I need assistance with an intricate question that I need dire attention in. The question is:

A cubic polynomial gives remainders (5x+4)and (12x-1) when divided by x^2-x+2 and x^2+x-1 respectively. Find the polynomial.

I did long division and came up with 4 equations:

-a+b+c = 5

-2a-2b+d = 4

2a-b+c = 12

-a+b+d = -1

The answer is x^3-2x^2+8x+2, but I would like to know how to carry out the simultaneous equations as I'm struggling with it and getting different answers each time. It would be helpful to get a step by step look into solving these equations to find a, b, c and d.

Thank you so much.

ANSWER: The system of equations can be represented in matrix form:

https://www.flickr.com/photos/dwread/19931754430/

Then use Gaussian Elimination to put the matrix into row echelon form and solve the system:

https://www.flickr.com/photos/dwread/20111648542/in/dateposted-public/

---------- FOLLOW-UP ----------

QUESTION: I'm very confused with the Gaussian Elimination. So far, I could keep up with the 3rd step of the Gaussian Elimination, but a detailed insight into the 4th step, a step by step format in working the solution, how d is 2 and not 12/11 according to the last step of the Gaussian Elimination would very much help me.

Thank you.

ANSWER: The 12 should be a 22 (because -6×10 + 82 = 22). Thus, 11d = 22, and d = 2.

Sorry about the mistake. It wouldn't have happened if I had gotten around to making an Excel spreadsheet for Gaussian Elimination. It is a straightforward algorithm, but with so many terms to be multiplied and added together, it is easy to make typographical or arithmetic errors.

Gauss-Jordan Elimination calls for even more calculations!—so that was the spreadsheet I chose to make.

https://www.flickr.com/photos/dwread/20102894196/in/dateposted-public/

---------- FOLLOW-UP ----------

QUESTION: I found a similar question that I seem to have a problem with. I tried what I learnt from you on the first problem on my own and was able to solve it with no problem, however, I had difficulty applying it to this question. I'm very sorry for taking your time.

Q. A cubic polynomial gives remainder (13x - 2) and (-1 - 7x) when divided by x^2 - x - 3 and x^ - 2x + 5 respectively. Find the polynomial.

First let's compare our systems of equations.

https://www.flickr.com/photos/dwread/19982288808/

-a+2b+c = -7

-10a-5b+d = -1

https://www.flickr.com/photos/dwread/19982288838/

4a+b+c = 13

3a+3b+d = -2

Did you get the same equations?

Augmented matrix of coefficients:

https://www.flickr.com/photos/dwread/20144162266/

(I rearranged the order of the equations)

I prefer Gauss-Jordan Elimination:

https://www.flickr.com/photos/dwread/20151361936/

The cubic polynomial: 3ax³ - 5x² + 6x + 4.

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