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

Solution

No, x2 – 1 cannot be the quotient on division of x6 + 2x3 + x – 1 by a polynomial in x of degree 5.

Explanation:

When a degree 6 polynomial is divided by degree 5 polynomial,

The quotient will be of degree 1.

Let us assume hat (x2 – 1) divides the degree 6 polynomial with and the quotient obtained is degree 5 polynomial (1)

As per 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,we get to know 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.

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