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Question

# Question 1 (i) Answer the following and justify Can x2–1 be the quotient on division of x6+2x3+x−1 by a polynomial in x of degree 5?

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Solution

## (i) No, because whenever we divide a polynomial x6+2x3+x–1 by a polynomial in x of degree 5, then we get quotient always as in linear form i.e., polynomial in x of degree 1. Let division = a polynomial in x of degree 5 = ax5+bx3+cx3+dx2+ex+1 Quotient = x2−1 And dividend = x6+2x3+x–1 By division algorithm for polynomials Dividend = divisor × quotient + remainder = (ax5+bx4+cx3+dx2+ex+f)×(x2−1)+ remainder = (a polynomial of degree 7) + remainder [in division algorithm, degree of divisor > degree of remainder] = (a polynomial of degree 7) But dividend = a polynomial of degree 6 So, division algorithm is not satisfied Hence, x2 – 1 is not a required quotient.

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