The product of two numbers is 16x⁴ -1. If one number is 2x - 1, find the other.

The product = (16x⁴ -1)

One number = (2x - 1)

Let the other number be A.

(2x - 1) $\times$ A = (16x⁴ -1)

⇒ A = $\dfrac{(16x^4 - 1)}{(2x - 1)}$

⇒ A = $\dfrac{(4x^2 - 1)(4x^2 + 1)}{(2x - 1)}$

⇒ A = $\dfrac{(2x - 1)(2x + 1)(4x^2 + 1)}{(2 - 1)}$

⇒ A = $\dfrac{\cancel{(2x - 1)}(2x + 1)(4x^2 + 1)}{\cancel{(2x - 1)}}$

⇒ A = (2x + 1)(4x² + 1)

⇒ A = 2x(4x² + 1) + 1(4x² + 1)

⇒ A = 8x⁽²⁺¹⁾ + 2x + 4x² + 1

⇒ A = 8x³ + 4x² + 2x + 1

Hence, the other number is 8x³ + 4x² + 2x + 1.