Factor each polynomial completely. See Example 6. 4m²p - 12mnp + 9n²p

Verified step by step guidance

1. Identify the greatest common factor (GCF) of all the terms in the polynomial \$4m^{2}p - 12mnp + 9n^{2}p$. Look for common variables and coefficients shared by each term.
2. Factor out the GCF from the polynomial. This will simplify the expression and make it easier to factor the remaining quadratic expression inside the parentheses.
3. Focus on the quadratic expression inside the parentheses after factoring out the GCF. Recognize that it is a trinomial of the form \(ax^{2} + bx + c\), where the variables are \(m\) and \(n\).
4. Use the method of factoring trinomials (such as the perfect square trinomial or the product-sum method) to factor the quadratic expression inside the parentheses completely.
5. Write the fully factored form by combining the GCF factored out initially with the factored form of the quadratic expression inside the parentheses.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process helps simplify expressions and solve equations. Common methods include factoring out the greatest common factor, grouping, and recognizing special products like perfect square trinomials.

Greatest Common Factor (GCF)

The GCF is the largest expression that divides all terms of a polynomial without leaving a remainder. Identifying the GCF is the first step in factoring, as it simplifies the polynomial and makes further factoring easier. For example, in 4m²p - 12mnp + 9n²p, the GCF includes variables and coefficients common to all terms.

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 a² ± 2ab + b² = (a ± b)². Recognizing this pattern allows quick factoring. In the given polynomial, after factoring out the GCF, the remaining trinomial may fit this pattern.