Problem 76 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$9 x^{3}-9 x$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(9x^{3}-9x\) is \(9x(x+1)(x-1)\)

Step 1: Identifying the Common Factor

The given polynomial is \(9x^{3}-9x\). Observe that both terms share a common factor of \(9x\). The first step in factoring a polynomial is to factor out any common factor.

Step 2: Apply Distributive Law to Factor Out the Common Factor

Factor out the common factor of \(9x\) from all terms of the polynomial. This is done by applying the distributive law in reverse. The polynomial becomes \(9x(x^{2}-1)\).

Step 3: Factor Completely

The polynomial inside the brackets can be further factored using the identity \(a^{2}-b^{2} = (a+b)(a-b)\). The polynomial becomes \(9x(x+1)(x-1)\).