Can x2 – 1 be the quotient on division of x6 + 2x3 + x – 1

1.	Can x2 – 1 be the quotient on division of x6 + 2x3 + x – 1 by a polynomial in x of degree 5?Justify your answer.
Answer» 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.

Can x2 – 1 be the quotient on division of x6 + 2x3 + x – 1 by a polynomial in x of degree 5?Justify your answer.

Answer»

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.