Problem 38 - A Graphical Approach to College Algebra

Question

Factor each perfect square trinomial completely. $$(a-3 b)^{2}-6(a-3 b)+9$$

Step-by-Step Solution

Answer

The expression factors into \((a - 3b - 3)^2\).

Step 1: Identify the Structure

Recognize that we are dealing with an expression that could fit the form of a perfect square trinomial. For this particular problem, notice that it looks like \((a-3b)^{2} - 6(a-3b) + 9\).

Step 2: Simplify the Expression

Let \(x = (a-3b)\). Substitute \(x\) into the expression to simplify it, so we have \(x^{2} - 6x + 9\). This is now a quadratic expression in terms of \(x\).

Step 3: Recognize the Perfect Square Form

Identify the expression \(x^{2} - 6x + 9\) as a perfect square trinomial. Recall that a perfect square trinomial has the form \((x-y)^2 = x^2 - 2yx + y^2\).

Step 4: Determine the Perfect Square

Since \(x^2 - 6x + 9\) is in the form \(x^2 - 2yx + y^2\), compare and determine that \(y = 3\) because \(-2y = -6\). Thus, \((x - 3)^2 = x^2 - 6x + 9\).

Step 5: Substitute Back to Original Variable

Substitute \(x = (a - 3b)\) back into the perfect square form we found: \((a - 3b - 3)^2\).

Step 6: Final Expression

Conclude that the original expression \((a - 3b)^2 - 6(a - 3b) + 9\) factors completely into \((a - 3b - 3)^2\).

Key Concepts

Factoring Quadratics

Factoring quadratics is an essential skill in algebra. It's the process of breaking down a quadratic expression into the product of two simpler expressions. The goal is to find values that, when multiplied together, recreate the original quadratic.

To begin factoring a quadratic, you should first recognize the standard form of the expression, which is usually written as \( ax^2 + bx + c \). In this particular exercise, we're dealing with a slightly modified form, but the underlying principles remain the same.

• Identify potential common factors or patterns, such as perfect squares.
• Determine values for variables like \(x\) that simplify the expression.
• Rewrite the quadratic in a factored form.

Here, the expression was simplified by substituting \(x = (a-3b)\) to easily recognize it as a perfect square trinomial, which is a special type of quadratic. This step helps in seeing the structure before factoring.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two. They typically involve terms with variables raised to the second power. The general form of a quadratic is \(ax^2 + bx + c\). Understanding this structure is crucial when simplifying and factoring such expressions.

Quadratic expressions can often take special forms that make them easier to work with:

• A perfect square trinomial, which can be factored into the square of a binomial.
• The difference of squares, where the quadratic splits into two conjugates.

In our exercise, the expression \((a-3b)^{2} - 6(a-3b) + 9\) was identified as resembling a perfect square trinomial once simplified to \(x^2 - 6x + 9\). Recognizing such patterns allows for effective factoring and simplification.

Polynomial Simplification

Simplifying polynomials involves reducing expressions by applying arithmetic operations and algebraic identities. The aim is to express the polynomial in its simplest form.

In the case of a quadratic expression, simplification might involve:

• Identifying common factors
• Reorganizing terms
• Using substitution for clearer visualization

For our given problem, simplifying meant recognizing the expression as a perfect square. By substituting \(x = (a-3b)\), the expression became \(x^2 - 6x + 9\), a known simplified form. Having identified this form, the next step was straightforward factoring into a perfect square: \((x - 3)^2\). Finally, substituting back to the original variable completed the simplification process.