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Please help me..

I have come across this type of problems many times in brilliant(sure you would to)

A monic polynomial of degree \(N\) leaves the remainder:-

\(A\) when divided by \((x \pm n_{1})\)

\(B\) when divided by \((x \pm n_{2})\)

\(C\) when divided by \((x \pm n_{3})\)

\(D\) when divided by \((x \pm n_{4})\)

\(E\) when divided by \((x \pm n_{5})\)

\(F\) when divided by \((x \pm n_{6})\)

and so on.....

Find out the value of \(P(Z)\)

So, I just wanted to know a more or less shortcut method to solve this type of problem rather than just to plug the values and solve the equations(this can sometimes become a hilarious job)..

Any type of help will be appreciated.

\(\color{ LimeGreen}{\text{Thank you}}\).

Note by Anik Mandal
2 years, 11 months ago

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1 vote

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Anik, because the polynomial problems are linear, we can always solve it using matrix. For example for the problem: Am I cubic?, my solution is as follows. I use an Microsoft Excel spreadsheet to do the matrix calculations.

Assuming that \(f(x)\) is a cubic polynomial, we can write in matrix form:

\(XA=B\quad \Rightarrow \begin{bmatrix} 1^3 & 1^2 & 1^1 & 1^0 \\ 2^3 & 2^2 & 2^1 & 2^0 \\ 3^3 & 3^2 & 3^1 & 3^0 \\ 4^3 & 4^2 & 4^1 & 4^0 \end{bmatrix} \begin{bmatrix} a_3 \\ a_2 \\ a_1 \\ a_0 \end{bmatrix} = \begin{bmatrix} 4 \\ 3 \\ 4 \\ 7 \end{bmatrix} \)

We can find \(A\) as follows:

\(A = X^{-1}B = \begin{bmatrix} -\frac{1}{6} & \frac {1}{2} &-\frac {1}{2} & \frac {1}{6} \\ \frac {3}{2} & -4 & \frac {7}{2} &-1 \\ -\frac {13}{3} & \frac {21}{2} & -7 & \frac{11}{6} \\ 4 & -6 & 4 & -1 \end{bmatrix} \begin{bmatrix} 4 \\ 3 \\ 4 \\ 7 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ -4 \\ 7 \end{bmatrix} \)

It is shown that: \(f(x) = x^2-4x+7 \) showing that: \(\boxed{No}\), it is not a cubic polynomial.

The spreadsheet.

Chew-Seong Cheong - 2 years, 10 months ago

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Sir how did we conclude that it is quadratic polynomial

Gaurav Jain - 2 years, 9 months ago

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Gaurav, \(A = \{0, 1, -4, 7\}\) means that \(f(x) = (0)x^3 + x^2-4x+7\).

Chew-Seong Cheong - 2 years, 9 months ago

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@Chew-Seong Cheong Oh! I got , Thank you Sir

Gaurav Jain - 2 years, 9 months ago

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Thanks@Chew-Seong Cheong

Anik Mandal - 2 years, 10 months ago

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If all the \(n\)'s are consecutive, you can use the Method of Differences. Check it's Wiki.

Satvik Golechha - 2 years, 11 months ago

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Thanks...

Anik Mandal - 2 years, 10 months ago

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@Aditya Raut @Satvik Golechha @Calvin Lin @Krishna Ar @Sandeep Bhardwaj @Trevor Arashiro @Finn Hulse @Agnishom Chattopadhyay @brian charlesworth and all other \(\color{ Red}{\text{Brilliant' ians}}\)

Anik Mandal - 2 years, 11 months ago

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Yes. Satvik is right. Method of differences is easy to employ too.

Krishna Ar - 2 years, 11 months ago

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