Problem 66 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$5 x^{3}-45 x$$

Step-by-Step Solution

Answer

The factored form of \(5x^{3} - 45x\) is \(5x(x+3)(x-3)\).

Step 1: Identify Common Factors

First, look at the terms in the polynomial and identify any common factors. Both terms, \(5x^{3}\) and \(-45x\), have a common factor of \(5x\).

Step 2: Factor Out the Common Factor

Next, divide every term in the polynomial by this common factor. This means dividing the term \(5x^{3}\) by \(5x\) to get \(x^{2}\), and \(-45x\) by \(5x\) to get \(-9\). This allows to rewrite the polynomial as \(5x(x^{2}-9)\).

Step 3: Check if Remaining Polynomial Can be Factored Further.

Then, check if the remaining polynomial inside the parentheses can be factored further. \(x^{2}-9\) can be factored into \((x+3)(x-3)\) because it's a difference of squares.

Step 4: Write Down the Final Factorized Form

Finally, substitute the factorized form \((x+3)(x-3)\) back into our factored polynomial. This gives the final fully factorized form of the polynomial as \(5x(x+3)(x-3)\).