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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 ----------

Gaussian Elimination step 4
Gaussian Elimination s  
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 -610 + 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.

Answer
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: 3ax - 5x + 6x + 4.

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Janet Yang

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I can answer questions in Algebra, Basic Math, Calculus, Differential Equations, Geometry, Number Theory, and Word Problems. I would not feel comfortable answering questions in Probability and Statistics or Topology because I have not studied these in depth.

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I tutor students (fifth through twelfth grades) and am a Top Contributor on Yahoo!Answers with over 24,000 math solutions.

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Co-author of An Outline of Scientific Writing: For Researchers With English as a Foreign Language

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I have a Bachelor's degree in Applied Mathematics from the University of California at Berkeley.

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George White Elementary School. Homework Help program at the Ridgewood Public Library, Ridgewood, NJ. Individual students.

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