Can x2 - 1 be the quotient on division of x6 + 2x3 + x - 1 by a polynomial in x of degree 5?


No, x2 - 1 cannot be the quotient on division of x6 + 2x3 + x - 1 by a polynomial in x of degree 5. This is because when we divide a degree 6 polynomial with degree 5 polynomial we get the quotient to be of degree 1.

Let if possible (x2 - 1) divides the polynomial with degree 6 and the quotient obtained is degree 5 polynomial (1)


i.e.,(degree6 polynomial) = (x2 - 1)(degree 5 polynomial) + r(x) (a = bq + r)


= (degree 7 polynomial) + r(x) (x2 term × x5 term = x7 term)


= (degree 7 polynomial)


Clearly, (degree6 polynomial) (degree 7 polynomial)


Hence, (1) is contradicted


x2 - 1 cannot be the quotient on division of x6 + 2x3 + x - 1 by a polynomial in x of degree 5


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