Question

Evaluate:

(i) (x - 2) (x + 2) (x² + 4)​

Answer 1

✑ Solution :

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☞ Distributive Property -

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• (i) (x - 2) (x + 2) (x² + 4)

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⇒ x(x + 2)(x² + 4) - 2(x + 2)(x² + 4)

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• Futher Distributing :

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⇒ x[x(x² + 4) + 2(x² + 4)] - 2[x(x² + 4) + 2(x² + 4)]

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⇒ x[ x³ + 4x + 2x² + 8] - 2[ x³ + 4x + 2x² + 8]

⇒ x⁴ + 4x² + 2x³ + 8x - 2x³ - 8x - 4x² - 16

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• Canceling the like terms :

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$\large⇒ \rm \: x⁴ + \cancel{ 4x²} + 2x³ + 8x \\ \large \rm- 2x³ - 8x \: \: \cancel{ - 4x²} - 16$

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$\large⇒ \rm \: x⁴ + 2x³ + \cancel{8x } \\ \large \rm- 2x³ \cancel{ - 8x} \: \:- 16$

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$\large⇒ \rm \: x⁴ + \cancel{2x³} \large \rm \: \: \: \cancel{- 2x³} - 16$

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$\large \therefore \rm(i) \: (x - 2) (x + 2) (x² + 4) \\ \\ = \large \: \boxed{ \underline\mathtt{ {x}^{4} - 16 }}$

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➢ Solving with Identity :

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• Identity :

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➥ (a + b)(a - b) = a² - b²

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(i) (x - 2) (x + 2) (x² + 4)

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❒ Here -

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• a = x

• b = 2

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⇒ (x - 2) (x + 2) (x² + 4) = (x² - 2²)(x² + 4)

⇒ (x² - 4)(x² + 4)

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• Again Using Identity :

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⇒ (x² - 4)(x² + 4)

$\large \rm \: = \: {x}^{ {2}^{2} } - {4}^{2} \\ \\ \large \rm \: = \: {x}^{2 \times 2} - 16$

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= x⁴ - 16

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$\large \therefore \rm(i) \: (x - 2) (x + 2) (x² + 4) \\ \\ = \large \: \boxed{ \underline\mathtt{ {x}^{4} - 16 }}$

Answer 2

$(x - 2)(x + 2)(x {}^{2} + 4) \\ = ( {x}^{2} - 4)( {x}^{2} + 4) \\ = {x}^{4} - 16$