Problem 75 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$x^{3}-4 x$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(x^{3}-4 x\) is \(x*(x-2)*(x+2)\).

Step 1: Identify the Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) for \(x^{3}-4 x\) is \(x\). This is because \(x\) is the common factor for both \(x^{3}\) and \(-4x\). So, we factor out \(x\) from \(x^{3}-4 x\), which gives us \(x*(x^{2}-4)\).

Step 2: Further Break Down the Resulting Factors

Now, we need to check if the remaining expression \(x^{2}-4\) can be further factored. In this case, \(x^{2}-4\) is a difference of squares, which is a common factoring pattern that can be written as \(a^{2} - b^{2} = (a - b)(a + b)\). Applying this rule to \(x^{2}-4\), we get \((x-2)(x+2)\).

Step 3: Present the final factored form

Upon replacing \(x^{2}-4\) with its factored form in \(x*(x^{2}-4)\), we get the completely factored form of the polynomial as \(x*(x-2)*(x+2)\).