Factor each trinomial, if possible. See Examples 3 and 4. (a-3b)^2-6(a-3b)+9

Verified step by step guidance

1. Recognize that the given expression \((a-3b)^2 - 6(a-3b) + 9\) is a quadratic trinomial in terms of \(u = a-3b\).
2. Rewrite the expression as \(u^2 - 6u + 9\).
3. Notice that this is a perfect square trinomial, which can be factored as \((u - 3)^2\).
4. Substitute back \(u = a-3b\) into the factored form to get \((a-3b - 3)^2\).
5. Simplify the expression to \((a-3b-3)^2\), which is the factored form of the original trinomial.

Key Concepts

Factoring Trinomials

Factoring trinomials involves rewriting a quadratic expression in the form ax^2 + bx + c as a product of two binomials. This process often requires identifying two numbers that multiply to ac (the product of a and c) and add to b. Understanding this concept is essential for simplifying expressions and solving equations.

Perfect Square Trinomials

A perfect square trinomial is a specific type of trinomial that can be expressed as the square of a binomial. The general form is (a ± b)² = a² ± 2ab + b². Recognizing this pattern helps in quickly factoring expressions like (a-3b)², which simplifies the factoring process.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique involves manipulating the expression to create a binomial square, which can then be factored easily. It is particularly useful for solving quadratic equations and understanding the properties of parabolas.