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Choose whether the statement is true or false:

$x^{2} - 1$ is the quotient on division of $x^{6} + {2 x}^{3} + x - 1$ by a polynomial in x of degree 5.

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True

False

answer is B.

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Detailed Solution

When a degree 6 polynomial is divided by degree 5 polynomial, The quotient will be of degree 1.
Assume that (

x^{2}

– 1) divides the degree 6 polynomial with and the quotient obtained is degree 5 polynomial (1) According to our assumption, (degree 6 polynomial) = (

x^{2}

– 1)(degree 5 polynomial) + r(x)
[ Since, (a = bq + r)] = (degree 7 polynomial) + r(x) [ Since, (

x^{2}

term ×

x^{5}

term =

x^{7}

term)] = (degree 7 polynomial) From the above equation, it is clear that our assumption is contradicted.

x^{2}

– 1 cannot be the quotient on division of

x^{6}

+ 2

x^{3}

+ x – 1 by a polynomial in x of degree 5.
So, the given statement is true.
Hence (2) is the correct option.

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