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 expression is a quadratic in terms of the binomial \( (a - 3b) \). Let \( x = (a - 3b) \) to simplify the expression.
2. Rewrite the expression using \( x \): \( x^2 - 6x + 9 \). This is now a standard quadratic trinomial.
3. Look for two numbers that multiply to the constant term (9) and add up to the coefficient of \( x \) (-6). These numbers are -3 and -3.
4. Use these numbers to factor the quadratic trinomial as \( (x - 3)(x - 3) \) or \( (x - 3)^2 \).
5. Substitute back \( x = (a - 3b) \) to write the factored form as \( ((a - 3b) - 3)^2 \), which simplifies to \( (a - 3b - 3)^2 \).

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials or other simpler expressions. This process helps simplify expressions and solve equations by finding roots or zeros of the polynomial.

Recognizing Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial, typically in the form a^2 ± 2ab + b^2. Identifying this pattern allows for quick factoring using the formula (a ± b)^2.

Substitution Method in Factoring

The substitution method involves temporarily replacing a complex expression with a single variable to simplify factoring. After factoring the simpler expression, the original terms are substituted back to complete the factorization.