Question

Can x^2-1 be the quotient on division of x^6+2x^3+x-1 by a polynomial in x of degree 5

Answer 1

$\bf\huge\underline{Question}$

Can x^2-1 be the quotient on division of x^6+2x^3+x-1 by a polynomial in x of degree 5

$\bf\huge\underline{Solution}$

Let divisor be a polynomial in x of degree 5 = ax^5 + bx⁴ + cx³ + dx² + ex + f

Quotient = x² - 1

and dividend = x^6 + 2x³ + x - 1

By division algorithm, we have

dividend = divisor × quotient + remainder

= (ax^5 + bx⁴ + cx³ + dx² + ex + f)×(x² - 1) ⠀+ remainder

= (a polynomial of degree 7)

But dividend is a polynomial of degree 6.

So, division algorithm is not satisfied.

Hence, x² - 1 can't be the quotient.