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factorize into as many factors as possible

To factorize the expression \( 6 \cdot b^4 - 6 \cdot b^3 - 72 \cdot b^2 \) into as many factors as possible, follow these steps:

1. **Factor out the greatest common factor (GCF):**
   - The GCF of the coefficients \(6\), \(-6\), and \(-72\) is \(6\).
   - The GCF of the variable terms \(b^4\), \(b^3\), and \(b^2\) is \(b^2\).

   So, factor out \(6b^2\) from the entire expression:

   \[
   6b^2(b^2 - b - 12)
   \]

2. **Factor the quadratic expression \(b^2 - b - 12\):**
   - We need to find two numbers that multiply to \(-12\) (the constant term) and add to \(-1\) (the coefficient of the linear term).

   The numbers \(-4\) and \(3\) work because:

   \[
   -4 \times 3 = -12 \quad \text{and} \quad -4 + 3 = -1
   \]

   So, factor the quadratic:

   \[
   b^2 - b - 12 = (b - 4)(b + 3)
   \]

3. **Combine the factors:**

   Substitute back into the expression:

   \[
   6b^2(b - 4)(b + 3)
   \]

Thus, the fully factorized form of the expression is:

\[
6b^2(b - 4)(b + 3)
\]

This is the factorized expression with as many factors as possible.

Would you like more details on any step? Here are some related questions you might explore:

1. How do you determine the greatest common factor (GCF) of a polynomial?
2. What is the process of factoring a quadratic expression?
3. How do you factor trinomials where the leading coefficient is not 1?
4. How can you use the distributive property to check your factorization?
5. What are some techniques for factoring more complex polynomials?

**Tip:** Always check your factorizations by expanding the factors back to ensure the original expression is obtained.

Learn how to factorize the expression 6b^2(b^2 - b - 12) into as many factors as possible with a detailed step-by-step guide. This problem covers key algebraic concepts such as factoring polynomials, finding the greatest common factor (GCF), and factoring quadratic expressions. Suitable for high school students (Grades 10-12).

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