Problem 68 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$6 x^{2}-18 x-60$$

Step-by-Step Solution

Answer

The factored form of the given polynomial is \(6 (x-5)(x+2)\).

Step 1: Identify Common Factors

The first step is to identify any common factors in all of the terms. The polynomial is \(6 x^{2}-18 x-60\). Inspecting the coefficients of each term reveals the common factor to be 6.

Step 2: Factor out the Common Factor

The next step is to factor out the common factor from all terms in the polynomial. Doing so gives: \(6 (x^{2}-3x-10)\).

Step 3: Factor the Quadratic

Then we factor the quadratic expression \(x^{2}-3x-10\). Factoring the quadratic will convert it into the form \(a(x-h)^{2}+k\), looking for two numbers that multiply to -10 and add to -3, -5 and 2 satisfy this condition, yielding: \(6 (x-5)(x+2)\).