Problem 47 - Algebra and Trigonometry

Question

Factor the difference of two squares. $$16 x^{4}-81$$

Step-by-Step Solution

Answer

The factorized form of the expression \(16x^4 - 81\) is \((2x - 3)(2x + 3)(4x^2 + 9)\).

Step 1: Identifying the components

Identify the squares in the expression. Here, the squares are \(a = 4x^2\) (since \(4x^2 * 4x^2 =16x^4\)) and \(b = 9\) (since \(9 * 9 = 81\)).

Step 2: Factorizing

Factorize the expression \(16x^4 - 81\) with the formula \(a^2 - b^2 = (a - b)(a + b)\). Here, substitute \(a = 4x^2\) and \(b = 9\) into the formula. You get \((4x^2 - 9)(4x^2 + 9)\).

Step 3: Further factorize

The term \((4x^2 - 9)\) can be further factorized because it's another difference of squares. Here, \(a = 2x\) (since \(2x * 2x =4x^2\)) and \(b = 3\) (since \(3 * 3 = 9\)). Apply the formula to get \((2x - 3)(2x + 3)(4x^2 + 9)\)