If ${(5 x^{2} + 14 x + 2)}^{2} - {(4 x^{2} - 5 x + 7)}^{2}$ is divided by $x^{2} + x + 1$ , then the quotient $q$ and the remainder $r$ are given by:

q = (x^{2} + 19 x - 5), r = 1

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q = 9 (x^{2} + 19 x - 5), r = 0

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q = (x^{2} + 19 x - 5), r = 0

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q = 9 (x^{2} + 19 x - 5), r = 1

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Solution

The correct option is B $q = 9 (x^{2} + 19 x - 5), r = 0$
As given ${(5 x^{2} + 14 x + 2)}^{2} - {(4 x^{2} - 5 x + 7)}^{2}$ is divided by $x^{2} + x + 1$
$= ((5 x^{2} + 14 x - 5) - (4 x^{2} - 5 x + 7)) ((5 x^{2} + 14 x + 2) + (4 x^{2} - 5 x + 7))$
$= (5 x^{2} + 14 x + 2 - 4 x^{2} + 5 x - 7) (5 x^{2} + 14 x + 2 + 4 x^{2} - 5 x + 7)$
$= (x^{2} + 19 x - 5) (9 x^{2} + 9 x + 9)$
$= (x^{2} + 19 x - 5) 9 (x^{2} + x + 1)$
Now dividing by $x^{2} + x + 1$
$= \frac{(x^{2} + 19 x - 5) 9 (x^{2} + x + 1)}{x^{2} + x + 1}$
$Q u o t i e n t = 9 (x^{2} + 19 x - 5) a n d R e m i n d e r = 0$

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Similar questions

If $({5 x}^{2} + 14 x + 2)^{2}$ – $({4 x}^{2} - 5 x + 7)^{2}$ is divided by $(x^{2} + x + 1)$ , then quotient q and remainder r are given by:

(5 x^{2} + 14 x + 2)^{2} - (4 x^{2} - 5 x + 7)^{2}

(x^{2} + x + 1)

q

r

(5 x^{2} + 14 x + 2)^{2} - (4 x^{2} - 5 x + 7)^{2}

x^{2} + x + 1

7 x^{5} - 2 x^{4} - 5 x^{3} + x^{2} - 3 x + 5

x - 2

x + y = 6, 2 x + y = 4

x + 2 y = 5

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