Problem 9 - Algebra and Trigonometry

Question

Factor out the greatest common factor. $$x^{2}(x-3)+12(x-3)$$

Step-by-Step Solution

Answer

The expression with the greatest common factor factored out is \((x - 3)(x^{2} + 12)\).

Step 1: Identify the Common Factor

The first step is to identify the greatest common factor that is shared amongst all the terms in the expression. In this case, both terms of the expression \(x^{2}(x-3)+12(x-3)\) share the common factor \(x - 3\).

Step 2: Factor out the Greatest Common Factor

The next step is to factor the common element out of the expression. This is done by dividing each term in the expression with the common factor and writing the result outside the brackets together with the common factor. Applying this to the expression gives us a new expression: \((x - 3)(x^{2} + 12)\)

Step 3: Check the Factoring

The final step is to verify the factoring by multiplying back the factored expression and comparing it with the original expression. Multiply \((x - 3)\) with both terms inside the brackets and compare it with the original expression: \(x^{2}(x-3)+12(x-3)\). If it gives the original expression back, then factoring was done correctly.