Problem 75 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: Searching for the greatest common factor

The given polynomial is \(x^{3}-4x\). Start by identifying the common factors. We can see that 'x' can be factored out from all the terms of the expression.

Step 2: Factoring out the common factor

By factoring the common factor 'x' out of each term, the given polynomial is rewritten as \(x(x^{2}-4)\).

Step 3: Recognizing difference of squares

The expression inside the brackets, \(x^{2}-4\), is a difference of squares which can further be factored. A difference of squares is an expression of the form \(a^{2}-b^{2}\), and it can be factored into \((a+b)(a-b)\).

Step 4: Factoring the difference of squares

Factoring the expression \(x^{2}-4\) using the formula for difference of squares, it becomes \((x+2)(x-2)\).

Step 5: Final factored form of the polynomial

Combining all the factored expressions together, the final factored form of the polynomial \(x^{3}-4x\) is \(x(x+2)(x-2)\).