Question

Factories the following:

(i) x^2 + 1/x^2 - 2 - 3x + 3/x.
(ii) a^2 x^2 + (ax^2 + 1) x + a.

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Answer 1

$\large\underline{\sf{Solution-(i)}}$

$\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 - 3x + \dfrac{3}{x}$

can be rewritten as

$\rm \:  =  \:\: {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 \times x \times \dfrac{1}{x} - 3x + \dfrac{3}{x}$

We know that,

$\boxed{ \bf{ \: {x}^{2} + {y}^{2} - 2xy = {(x - y)}^{2} }}$

$\rm \:  =  \:  \: {\bigg(x - \dfrac{1}{x} \bigg) }^{2} - 3\bigg(x - \dfrac{1}{x} \bigg)$

$\rm \:  =  \:  \: \bigg(x - \dfrac{1}{x} \bigg)\bigg(x - \dfrac{1}{x} - 3 \bigg)$

$\large\underline{\sf{Solution-(ii)}}$

$\rm :\longmapsto\: {a}^{2} {x}^{2} + ( {ax}^{2} + 1)x + a$

can be rewritten as

$\rm \:  =  \:  \: {a}^{2} {x}^{2} + {a}^{} {x}^{3} + x + a$

$\rm \:  =  \:  \: {a} {x}^{2}(a + x) + 1(a + x)$

$\rm \:  =  \:  \: (x + a)( {ax}^{2} + 1)$

Additional Information : -

More Identities to know :-

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

(a + b)² = (a - b)² + 4ab

(a - b)² = (a + b)² - 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - b³ - 3ab(a - b)

Answer 2

Answer:

$\underline{\purple{\ddot{\Maths dude}}}$

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