No, x2 – 1 cannot be the quotient on division of x6 + 2x3 + x – 1 by a polynomial in x of degree 5.
Justification:
When a degree 6 polynomial is divided by degree 5 polynomial,
The quotient will be of degree 1.
Assume that (x2 – 1) divides the degree 6 polynomial with and the quotient obtained is degree 5 polynomial (1)
According to our assumption,
(degree 6 polynomial) = (x2 – 1)(degree 5 polynomial) + r(x) [ Since, (a = bq + r)]
= (degree 7 polynomial) + r(x) [ Since, (x2 term × x5 term = x7 term)]
= (degree 7 polynomial)
From the above equation, it is clear that, our assumption is contradicted.
x2 – 1 cannot be the quotient on division of x6 + 2x3 + x – 1 by a polynomial in x of degree 5
Hence Proved.
