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Chapter 7: Problem 146

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

Short Answer

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

Step by step solution

01

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:
02

Factor out the GCF

Factor 'q' out of each term to write the polynomial as a product. This gives: q(q^{2} - 5q - 24)
03

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)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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