Question

justify your answer can(x^2-1)be the quotient of division ofx^6+2x^3+x-1 by a polynomial in x of degree 5

Answer 1

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 divisor = a polynomial in x of degree 5
= ax5 + bx4 + cx3 + dx2 + ex + f
quotient = x2 -1
and dividend = x6 + 2x3 + x -1
By division algorithm for polynomials,
Dividend = Divisor x Quotient + Remainder
= (ax5 + bx4 + cx3 + dx2 + ex + f)x(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.