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Let's factor the given expression \( y^3 - 9y^2 + y - 9 \) by grouping.

### Steps:

1. **Group terms**:  
   Group the terms in pairs:
   \[
   (y^3 - 9y^2) + (y - 9)
   \]

2. **Factor each group**:  
   - From the first group \( y^3 - 9y^2 \), we can factor out \( y^2 \):
     \[
     y^2(y - 9)
     \]
   - From the second group \( y - 9 \), we can factor out 1 (since \( y - 9 \) cannot be further factored):
     \[
     1(y - 9)
     \]

3. **Factor out the common binomial**:  
   Both groups now have the common factor \( (y - 9) \), so we can factor that out:
   \[
   (y^2 + 1)(y - 9)
   \]

Thus, the factored form of \( y^3 - 9y^2 + y - 9 \) is:
\[
(y^2 + 1)(y - 9)
\]

Do you want further clarification or have any other questions?

### Related questions:
1. How do you apply grouping to more complex polynomials?
2. Can all cubic polynomials be factored by grouping?
3. What if the common factor is not immediately obvious—how do you handle that?
4. How does the factoring process change for higher-degree polynomials?
5. What other methods can be used if grouping doesn’t work?

### Tip:  
Always check if factoring can be done further or if a different technique like synthetic division might help when grouping isn't straightforward.

Learn how to factor the cubic polynomial y^3 - 9y^2 + y - 9 by grouping. This solution covers algebraic concepts of factoring polynomials step-by-step. Ideal for students in Grades 9-12.

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