Question

Simplify : (2x − 1)(2x + 1)(4x² + 1)(16x⁴ + 1)

Answer 1

To simplify, we will proceed as follows:

\[ \left( 2x - 1 \right)\left( 2x + 1 \right)\left( 4 x^2 + 1 \right)\left( 16 x^4 + 1 \right)\]

\[ = \left( \left( 2x \right)^2 - 1^2 \right)\left( 4 x^2 + 1 \right)\left( 16 x^4 + 1 \right) \left[ \because \left( a + b \right)\left( a - b \right) = a^2 - b^2 \right] \]

\[ = \left( 4 x^2 - 1 \right)\left( 4 x^2 + 1 \right)\left( 16 x^4 + 1 \right) \]

\[ = \left\{ \left( 4 x^2 \right)^2 - \left( 1^2 \right)^2 \right\}\left( 16 x^4 + 1 \right) \left[ \because \left( a + b \right)\left( a - b \right) = a^2 - b^2 \right]\]

\[ = \left( 16 x^4 - 1 \right) \left( 16 x^4 + 1 \right) \]

\[ = \left( 16 x^4 \right)^2 - 1^2 \left[ \because \left( a + b \right)\left( a - b \right) = a^2 - b^2 \right]\]

\[ = 256 x^8 - 1\]