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The given expression is \( y^2 - 9y + 18 \). We are tasked with factoring it completely or identifying it as a prime polynomial if it cannot be factored.

### Step 1: Check for factorization
We have a quadratic expression of the form \( y^2 + by + c \), where \( b = -9 \) and \( c = 18 \).

We need to find two numbers that multiply to give 18 (the constant term \( c \)) and add up to -9 (the coefficient of \( y \), which is \( b \)).

- The factors of 18 are: \( 1 \times 18 \), \( 2 \times 9 \), and \( 3 \times 6 \).
- However, no combination of these factors will add up to -9. For example:
  - \( 3 + 6 = 9 \), but not -9.

### Step 2: Conclusion
Since no integer factor pairs satisfy both conditions, the polynomial **cannot be factored** further using real numbers. Therefore, it is **prime**.

Would you like more details, or do you have any questions about this?

Here are 5 related questions to explore:
1. What is the general form for factoring a quadratic equation?
2. Can you factor a quadratic polynomial with irrational or complex roots?
3. How can you verify that a polynomial is prime?
4. What are other methods for factoring quadratics (e.g., completing the square)?
5. How does the discriminant help determine if a quadratic can be factored?

**Tip:** When factoring, always check both addition and multiplication conditions for finding factor pairs.

Factor completely. Select 'Prime' if the polynomial cannot be factored.

y^2 - 9y + 18

Learn how to factor the quadratic polynomial y^2 - 9y + 18 completely or identify it as prime. This problem covers core algebraic concepts, the standard quadratic equation formula, and factorization techniques. Suitable for students in Grades 8-10.

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