5r^6s^4t^5-15r^2s^7t^2-30r^3s^2t^4 factor

Let's factor the expression \(5r^6s^4t^5 - 15r^2s^7t^2 - 30r^3s^2t^4\) step by step.

### Step 1: Identify the Greatest Common Factor (GCF)
- **Coefficients**: The GCF of 5, 15, and 30 is 5.
- **Variable \(r\)**: The lowest power of \(r\) among the terms is \(r^2\).
- **Variable \(s\)**: The lowest power of \(s\) among the terms is \(s^2\).
- **Variable \(t\)**: The lowest power of \(t\) among the terms is \(t^2\).

So, the GCF of the entire expression is \(5r^2s^2t^2\).

### Step 2: Factor out the GCF
Now, we factor out the GCF from each term:

\[
5r^6s^4t^5 - 15r^2s^7t^2 - 30r^3s^2t^4 = 5r^2s^2t^2\left(\frac{r^6s^4t^5}{r^2s^2t^2} - \frac{15r^2s^7t^2}{5r^2s^2t^2} - \frac{30r^3s^2t^4}{5r^2s^2t^2}\right)
\]

### Step 3: Simplify the expression inside the parentheses
Let's simplify each term inside the parentheses:

- **First term**: \(\frac{r^6s^4t^5}{r^2s^2t^2} = r^{6-2}s^{4-2}t^{5-2} = r^4s^2t^3\)
- **Second term**: \(\frac{15r^2s^7t^2}{5r^2s^2t^2} = 3s^{7-2} = 3s^5\)
- **Third term**: \(\frac{30r^3s^2t^4}{5r^2s^2t^2} = 6r^{3-2}t^{4-2} = 6r t^2\)

Now, the expression becomes:

\[
5r^2s^2t^2(r^4s^2t^3 - 3s^5 - 6rt^2)
\]

### Final Factored Form
The completely factored form of the expression is:

\[
\boxed{5r^2s^2t^2(r^4s^2t^3 - 3s^5 - 6rt^2)}
\]

Would you like to explore any part of this solution further, or do you have other questions?

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Here are 5 related questions you might consider:

1. How would you factor the expression \(r^4s^2t^3 - 3s^5 - 6rt^2\) further, if possible?
2. What is the importance of factoring in simplifying algebraic expressions?
3. How do you factor polynomials that involve fractional exponents?
4. Can you use factoring to solve equations? If so, how?
5. How would you apply factoring in calculus, for example in finding critical points?

**Tip:** When factoring complex expressions, always check each term to ensure you’ve factored out the greatest common factor (GCF) completely. This can make further factoring or simplifying easier.

Explore the factorization process of the polynomial expression 5r^6s^4t^5 - 15r^2s^7t^2 - 30r^3s^2t^4 step-by-step. Learn about identifying the Greatest Common Factor (GCF), simplifying algebraic expressions, and factoring polynomials. Suitable for Grades 9-12.

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