How do you determine if a polynomial is a perfect square trinomial?

To determine if a polynomial is a perfect square trinomial, you need to check if it can be expressed in the form \((ax + b)^2\) or \((ax - b)^2\). A perfect square trinomial typically has the form:

\[a^2x^2 + 2abx + b^2\]

Here’s the step-by-step process to determine if a given trinomial is a perfect square:

1. **Identify the coefficients**:
   - Given a trinomial \(Ax^2 + Bx + C\), identify the coefficients \(A\), \(B\), and \(C\).

2. **Check the first term**:
   - Ensure the first term \(Ax^2\) is a perfect square. This means \(A\) should be a perfect square, say \(A = a^2\).

3. **Check the last term**:
   - Ensure the last term \(C\) is a perfect square. This means \(C\) should be a perfect square, say \(C = b^2\).

4. **Check the middle term**:
   - Check if the middle term \(B\) is twice the product of the square roots of the first and last terms. Specifically, check if \(B = 2ab\).

If all these conditions are satisfied, then the polynomial is a perfect square trinomial.

### Example:

Determine if the trinomial \(9x^2 + 12x + 4\) is a perfect square.

1. **Identify the coefficients**:
   - \(A = 9\), \(B = 12\), \(C = 4\).

2. **Check the first term**:
   - The first term is \(9x^2\), and \(9\) is a perfect square (\(9 = 3^2\)).

3. **Check the last term**:
   - The last term is \(4\), and \(4\) is a perfect square (\(4 = 2^2\)).

4. **Check the middle term**:
   - The middle term is \(12x\), and we need to check if \(12 = 2 \cdot 3 \cdot 2\). Indeed, \(12 = 2 \cdot 3 \cdot 2\).

Since all conditions are satisfied, \(9x^2 + 12x + 4\) is a perfect square trinomial and can be written as:

\[(3x + 2)^2\]

### Summary

To determine if a polynomial is a perfect square trinomial:
1. Verify that the first term is a perfect square.
2. Verify that the last term is a perfect square.
3. Ensure the middle term is twice the product of the square roots of the first and last terms.

Would you like more details or have any questions on this topic?

Here are 8 related questions you might find interesting:
1. How do you factor a perfect square trinomial?
2. What are some examples of perfect square trinomials in real-world applications?
3. How can you determine if a polynomial is factorable?
4. What is the difference between a perfect square trinomial and a general trinomial?
5. How do you solve quadratic equations using perfect square trinomials?
6. What are some common mistakes when identifying perfect square trinomials?
7. Can a perfect square trinomial have negative coefficients?
8. How do you complete the square to convert a quadratic equation into a perfect square trinomial?

**Tip:** When identifying perfect square trinomials, always double-check the coefficients to ensure accuracy in the verification process.

Discover how to determine if a polynomial is a perfect square trinomial with clear step-by-step instructions. This problem involves identifying coefficients, checking for perfect squares, and verifying the form (ax ± b)^2. Suitable for high school students studying polynomials and algebra.

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