Problem 39 - Algebra 1: Concepts and Skills

Question

Factor the expression completely. \(4 t^{3}-144 t\)

Step-by-Step Solution

Answer

The factored form of the expression \(4 t^{3} - 144 t\) is \(4t(t-6)(t+6)\)

Step 1: Identifying the Greatest Common Factor (GCF)

First, identify the greatest common factor of the terms \(4 t^{3}\) and \(-144 t\). Both terms are divisible by \(4t\), thus, \(4t\) is the GCF.

Step 2: Factoring out the GCF

Next, factor out the universal greatest common factor which is \(4t\). This leaves \(4 t^{3} - 144t = 4t(t^{2} - 36)\)

Step 3: Factoring the Difference of Two Squares

The expression inside the parentheses (i.e., \(t^{2} - 36\)) can be identified as a difference of squares, which can be factored further. The difference of squares follows the pattern \(a^{2} - b^{2} = (a - b)(a + b)\). Here, \(a = t\) and \(b = 6\). Factoring \(t^{2}-36\) yields \((t-6)(t+6)\)

Step 4: Final Factored Form

Rewrite the original equation using the factors you found: \(4t(t^{2}-36) = 4t(t-6)(t+6)\)