Let's simplify the expression step by step:

We are asked to simplify:

\[
\frac{36(y^4 - 5y^3 - 24y^2)}{9y(y-8)}
\]

### Step 1: Factor the numerator and denominator.
- The numerator is \(36(y^4 - 5y^3 - 24y^2)\).
  We can first factor out \(36y^2\) from the expression:
  
  \[
  36y^2(y^2 - 5y - 24)
  \]

- The denominator is \(9y(y - 8)\).
  
### Step 2: Simplify the constants.
- \(36\) divided by \(9\) is \(4\).

So, the simplified expression becomes:

\[
\frac{36y^2(y^2 - 5y - 24)}{9y(y - 8)} = 4 \cdot \frac{y^2(y^2 - 5y - 24)}{y(y - 8)}
\]

### Step 3: Cancel common factors.
- We have \(y^2\) in the numerator and \(y\) in the denominator, so we can cancel one factor of \(y\) from both the numerator and the denominator:

\[
4 \cdot \frac{y(y^2 - 5y - 24)}{(y - 8)}
\]

### Step 4: Factor the quadratic in the numerator.
Now we factor \(y^2 - 5y - 24\). We are looking for two numbers that multiply to \(-24\) and add to \(-5\). These numbers are \(-8\) and \(3\).

So, we can factor \(y^2 - 5y - 24\) as:

\[
y^2 - 5y - 24 = (y - 8)(y + 3)
\]

### Step 5: Simplify the expression.
Substituting this factorization back into the expression:

\[
4 \cdot \frac{y(y - 8)(y + 3)}{(y - 8)}
\]

Now we can cancel out the common factor \((y - 8)\) from the numerator and the denominator:

\[
4 \cdot y(y + 3)
\]

### Final Answer:
The simplified expression is:

\[
4y(y + 3)
\]

Would you like further clarification or details on any of the steps?

Here are five related questions you could explore:
1. How do you factor quadratic expressions efficiently?
2. What happens when there are no common factors to cancel out in such expressions?
3. Can you apply this process to expressions involving higher powers of \(y\)?
4. How does the distributive property work in simplifying algebraic expressions?
5. How can common factoring mistakes be avoided?

**Tip**: Always check for common factors between numerator and denominator to simplify expressions early in the process.

Learn how to simplify the rational expression 36(y^4 - 5y^3 - 24y^2) ÷ 9y(y-8) step-by-step. This detailed solution involves factoring and cancelling common factors, perfect for students in Grades 9-11 studying algebra and rational expressions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation