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