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Question
Simplify : (2x − 1)(2x + 1)(4x2 + 1)(16x4 + 1)
Solution
To simplify, we will proceed as follows:
\[ \left( 2x - 1 \right)\left( 2x + 1 \right)\left( 4 x^2 + 1 \right)\left( 16 x^4 + 1 \right)\]
\[ = \left( \left( 2x \right)^2 - 1^2 \right)\left( 4 x^2 + 1 \right)\left( 16 x^4 + 1 \right) \left[ \because \left( a + b \right)\left( a - b \right) = a^2 - b^2 \right] \]
\[ = \left( 4 x^2 - 1 \right)\left( 4 x^2 + 1 \right)\left( 16 x^4 + 1 \right) \]
\[ = \left\{ \left( 4 x^2 \right)^2 - \left( 1^2 \right)^2 \right\}\left( 16 x^4 + 1 \right) \left[ \because \left( a + b \right)\left( a - b \right) = a^2 - b^2 \right]\]
\[ = \left( 16 x^4 - 1 \right) \left( 16 x^4 + 1 \right) \]
\[ = \left( 16 x^4 \right)^2 - 1^2 \left[ \because \left( a + b \right)\left( a - b \right) = a^2 - b^2 \right]\]
\[ = 256 x^8 - 1\]
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