Factor the​ trinomial, or state that the trinomial is prime. Check the factorization using FOIL multiplication.
9 y squared minus 24 y plus 169y2−24y+16
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Part 1
Select the correct choice below and fill in any answer boxes within your choice.
A.9 y squared minus 24 y plus 16 equals enter your response here
9 y squared minus 24 y plus 169y2−24y+16equals=enter your response here
B.
The polynomial is prime.

To factor the trinomial \(9y^2 - 24y + 16\), we can follow these steps:

### Step 1: Identify the coefficients
- The trinomial is in the form \(ax^2 + bx + c\), where:
  - \(a = 9\)
  - \(b = -24\)
  - \(c = 16\)

### Step 2: Factor the trinomial
We can look for factors of the form \((my + n)(py + q)\) such that:
- The product of the first terms gives \(9y^2\).
- The product of the last terms gives \(16\).
- The middle term, \(-24y\), is obtained by adding the products of the outer and inner terms.

Let's try:
\[
(3y - 4)(3y - 4)
\]

### Step 3: Check the factorization using FOIL
Using the FOIL method:
- **First:** \(3y \times 3y = 9y^2\)
- **Outer:** \(3y \times -4 = -12y\)
- **Inner:** \(-4 \times 3y = -12y\)
- **Last:** \(-4 \times -4 = 16\)

Combining the middle terms:
\[
9y^2 - 12y - 12y + 16 = 9y^2 - 24y + 16
\]
The factorization is correct.

### Final Answer:
\[
9y^2 - 24y + 16 = (3y - 4)(3y - 4) = (3y - 4)^2
\]

This trinomial factors as \((3y - 4)^2\).

Would you like more details or have any questions?

Here are five related questions:
1. How do you factor trinomials when the leading coefficient is not 1?
2. What is the difference between a perfect square trinomial and a general trinomial?
3. Can all trinomials be factored into binomials?
4. How do you identify if a trinomial is a perfect square?
5. What are the steps to check factorization using the FOIL method?

**Tip:** When factoring, always check if the trinomial is a perfect square by comparing the middle term to twice the product of the square roots of the first and last terms.

Learn how to factor the trinomial 9y^2 - 24y + 16 using the FOIL method with a detailed step-by-step solution. This problem involves algebraic factorization and verification using FOIL multiplication. Suitable for students in Grades 9-12.

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