Problem 146 - Elementary Algebra

Question

Factor completely. \(q^{3}-5 q^{2}-24 q\)

Step-by-Step Solution

Answer

'q(q - 8)(q + 3)'

Step 1: Identify the Greatest Common Factor (GCF)

First, observe the polynomial to identify any common factors in each term. Here, each term shares a factor of 'q'. Hence, the GCF is 'q'. Now factor out 'q' from each term:

Step 2: Factor out the GCF

Factor 'q' out of each term to write the polynomial as a product. This gives: q(q^{2} - 5q - 24)

Step 3: Factor the Quadratic Expression

Next, focus on the quadratic expression inside the parentheses: q^{2} - 5q - 24. Look for two numbers that multiply to -24 and add to -5. These numbers are -8 and 3. Therefore, rewrite the quadratic expression as: q(q - 8)(q + 3)

Key Concepts

Greatest Common Factor

Now, we have a simpler expression to work with inside the parentheses.

Quadratic Expression

Now, the quadratic expression is fully factored.

Polynomial Factorization

Understanding polynomial factorization is crucial as it simplifies complex polynomials and is foundational for solving polynomial equations.