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State whether the following statement is true or false.

$x 2 − 1$ be 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

Given polynomial is,

x 6 + 2 x 3 + x − 1

Given polynomial is divided by a polynomial in

x

of degree 5.
According to division algorithm,
For any given polynomial

p (x)

and non-zero polynomial

g (x)

, there are polynomials

q (x)

and

r (x)

such that,

p (x) = g (x) q (x) + r (x)

, where

r (x) = 0

or degree

r (x) <

degree

g (x)

.
Here,

p (x) = x^{6} + 2 x^{3} + x - 1

g (x) = x

in degree 5,

q (x) = x^{2} - 1

so,

x^{6} + 2 x^{3} + x - 1 = (x

in degree 5 )

(x^{2} - 1) + r (x)

= x

in degree 7
Thus,

x^{2} - 1

can never be the quotient of

x^{6} + 2 x^{3} + x - 1

.
Therefore, given statement is false.
Hence, option (2) is correct.

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