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If P(x) =1+x+x^2+x^3+x^4+x^5 then find the remainder when P(x^12) is divided by P(x)

If P(x) =1+x+x^2+x^3+x^4+x^5 then find the remainder when P(x^12) is divided by P(x)

Grade:12

2 Answers

Nishant Vora IIT Patna
askIITians Faculty 2467 Points
8 years ago
Let me say x1, x2, x3,x4 and x5 are the roots of P(x)=0
Now roots will be

223-1903_Capture.PNG

Now If I divide P(x^12) by P(x) , remainder will be of deg=4
[this is a rule, deg of remainder is one less than deg of divisor]
So, P(x^12) = P(x)(quotient) + (ax^4 + bx^3 + cx^2 + dx^1 + e )

Now in above eqn there are 5 variables which can be found by puting x= x1, x2, x3,x4 and x5
and remember P(x1), P(x2), P(x3),P(x4) and P(x5) == 0

That’s it!!
mycroft holmes
272 Points
8 years ago
Adding some random words to make sure that the 100 characters minimum set for answers is achieved. The solution is below:

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