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Question

Question 1 (i)
Answer the following and justify
Can x21 be the quotient on division of x6+2x3+x1 by a polynomial in x of degree 5?

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Solution

(i) No, because whenever we divide a polynomial x6+2x3+x1 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 = x21
And dividend = x6+2x3+x1
By division algorithm for polynomials
Dividend = divisor × quotient + remainder
= (ax5+bx4+cx3+dx2+ex+f)×(x21)+ 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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