Problem 65 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The completely factored form of the polynomial is \(3x(x-1)(x+1)\)

Step 1: Identify common factors

The first step is to look at the polynomial and identify any common factors. In this case, both terms of \(3 x^{3} - 3x\) can be divided by '3x'.

Step 2: Factor out common factors

Once the common factors '3x' have been identified, they can be factored out from the polynomial. This is done by dividing each term by '3x', resulting in \(3x(x^{2} - 1)\).

Step 3: Factor the remaining polynomial

The remaining polynomial within the bracket \(x^{2} - 1\) is a difference of squares, which can also be factored. This is done using the formula for factoring a difference of squares, which is \(a^{2} - b^{2} = (a+b)(a-b)\). Using this formula, the \(x^{2} - 1\) can be written as \((x-1)(x+1)\).

Step 4: Combine your results

Now, combine the common factor '3x' factored out in step 2 with the result from step 3, resulting in the completely factored form of the given polynomial: \(3x(x-1)(x+1)\).