State whether the given statement is True or False.

Statement: Division algorithm can be verified, if $x^{3}$ + 1, x + 1, $x^{2}$ – x + 1 and 0 are dividend, divisor, quotient and remainder respectively.

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True

False

answer is A.

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Detailed Solution

It is given in the question that

x^{3}

+ 1, x + 1,

x^{2}

– x + 1 and 0 are dividend, divisor, quotient and remainder respectively.
Then, we have to verify the division algorithm.
Since, we have given that,
p(x) =

x^{3}

+ 1
q(x) = x + 1
d(x) =

x^{2}

– x + 1
r(x) = 0
The division algorithm states that for any polynomial p(x) and q(x) , there exists unique polynomial d(x) and r(x) such that
p(x) = q(x) × d(x) + r(x)
Now by taking L.H.S
p(x) =

x^{3}

+ 1
Now, by taking R.H.S
q(x) × d(x) + r(x)
∴ (x + 1) × (

x^{2}

– x + 1) + 0 ∴

x^{3}

−

x^{2}

+ x +

x^{2}

– x + 1
After cancellation the equal terms of opposite signs, we get

x^{3}

+ 1 which is same as p(x)
Hence, L.H.S = R.H.S Therefore, the division algorithm is verified.
So, option (1) is correct.
Hence, the given statement is true.

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