please solve the question only question number 8.
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Answers
Answer:
Remainder [r(x)] = 6
Step-by-step explanation:
As P(x) is the sum of G. P.
P(x) = (x^6 - 1) / (x - 1)
Now let us compute P(x^12) mod x^6 - 1.
This amounts replacing x^6 by 1 in all terms of P(x^12).
Sonce, it has 6 terms and all are the powers of x^6, we conclude that,
P(x^12) = 6 mod x^6 - 1
i. e.
P(x^12) = (x^6 - 1)•qx + 6
We know that, by Euclid's Division Algorithm,
Dividend = Divisor × Quotient + Remainder
=> f(x) = g(x) × q(x) + r(x)
Since, here P(x^12) is to be divided by P(x), then here g(x) = P(x).
So, f(x) = P(x^12).
P(x^12) = P(x)(x - 1)q(x) + 6
Then,
P(x^12) = P(x) • Q(x) + 6 , where Q = x-1.
Here, it came in its Euclid's Division Lemma form.
Clearly, we can see that, 6 is the remainder when P(x^12) is divided by P(x).
Remark - In general, you can always use this strategy with numbers, polynomials, or more general, in Euclidean rings. If you want to compute f mod g (the remainder),but you know how to calculate f mod h with h divisible by g, you can compute there remainder of division f by h and then reduce it to modulo g.
For example, if you know that,
x ≡ 5 mod 9, then you can conclude that its, x ≡ 2 mod 3, without even knowing the value of x.
NOTE : THE EXPLANATION OF THIS ANSWER IS CLEARLY GIVEN IN THE ATTACHMENT BELOW. KINDLY REFER TO THAT IN ORDER.
Answer:
As P(x) is the sum of GP. =
It has 5 roots , let a1, a2, a3, a4,a5, and they are the 6th roots of unity except unity.
Here R(x) is a remainder and a polynomial of maximum degree 4.
Put x=a 1 ,a 2 ...............,a 5
We get,R(a 1 )=6, R(a 2)=6 ,R(a 3 )=6, R(a 4)=6, R(a5 )=6
i.e, R(x)−6=0 has 6 roots.
Which contradict that R(x) is maximum of degree 4.
So, it is an identity
Therefore, R(x)=6.