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show that the polynomial x^4P +x^4Q+! +x^4R+2 +x4S+3 is divisible by x^3+x^2+x+1 where P,Q,R,S belongs to N

show that the polynomial  x^4P +x^4Q+! +x^4R+2 +x4S+3 is divisible by x^3+x^2+x+1 where P,Q,R,S belongs to N

Grade:12

1 Answers

Rajesh Juluru
13 Points
11 years ago

(x^4)-1=(x-1)(x^3+x^2+x+1)

So the roots of (x^3+x^2+x+1) are -1,i,-i

Let f(x)= x^4P +x^4Q+1 +x^4R+2 +x^4S+3

f(1)=0;f(i)=0;f(-i)=0

So,x^3+x^2+x+1 is a factor of the given polynomial

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