Factor each trinomial, if possible. See Examples 3 and 4. 9m²n²+12mn+4

Verified step by step guidance

1. Identify the trinomial to factor: \$9m^2n^2 + 12mn + 4$.
2. Check if the trinomial is a perfect square trinomial by examining the first and last terms: \$9m^2n^2\( is \)(3mn)^2\( and \(4\) is \)2^2$.
3. Check the middle term to see if it matches \(2 \times (3mn) \times 2 = 12mn\), which it does, confirming it is a perfect square trinomial.
4. Write the trinomial as the square of a binomial: \((3mn + 2)^2\).
5. Express the factored form clearly: \$9m^2n^2 + 12mn + 4 = (3mn + 2)^2$.

Key Concepts

Factoring Trinomials

Factoring trinomials involves rewriting a quadratic expression of the form ax^2 + bx + c as a product of two binomials. This process simplifies expressions and solves equations by finding values that satisfy the equation. Recognizing patterns and using methods like trial and error or the AC method helps in factoring.

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest expression that divides all terms of a polynomial without leaving a remainder. Identifying and factoring out the GCF simplifies the polynomial, making further factoring easier. Always check for a GCF before attempting other factoring methods.

Factoring Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, typically in the form a^2 + 2ab + b^2 = (a + b)^2. Recognizing this pattern allows quick factoring, especially when the first and last terms are perfect squares and the middle term is twice the product of their roots.