Problem 51 - Elementary and Intermediate Algebra

Question

Factor by first grouping the appropriate terms. \(x^{2}-2 x+1-9 z^{2}\)

Step-by-Step Solution

Answer

The factored form of the expression is \((x - 1 - 3z)(x - 1 + 3z)\).

Step 1: Identify Groupings

The given expression is \(x^{2} - 2x + 1 - 9z^{2}\). We need to determine how to group these terms to facilitate factoring. Notice that the first three terms \(x^2 - 2x + 1\) form a perfect square trinomial, which is a candidate for grouping.

Step 2: Factor the Perfect Square Trinomial

The expression \(x^2 - 2x + 1\) can be rewritten as \((x - 1)^2\) because it fits the formula of \((a - b)^2 = a^2 - 2ab + b^2\), with \(a = x\) and \(b = 1\). This makes our expression \((x - 1)^2 - 9z^2\).

Step 3: Recognize a Difference of Squares

The expression \((x - 1)^2 - 9z^2\) is a difference of squares. It fits the formula \(a^2 - b^2 = (a - b)(a + b)\), where \(a = x - 1\) and \(b = 3z\).

Step 4: Factor the Difference of Squares

Apply the difference of squares formula to \((x - 1)^2 - 9z^2\). It becomes \((x - 1 - 3z)(x - 1 + 3z)\).

Step 5: Verify the Factorization

Expand \((x - 1 - 3z)(x - 1 + 3z)\) to verify the factorization: You get \((x - 1)^2 - (3z)^2\), which simplifies back to the original expression \((x - 1)^2 - 9z^2 = x^2 - 2x + 1 - 9z^2\). Thus, the factorization is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a very useful concept in algebra when it comes to factoring polynomials. Recognizing one can significantly simplify your work. But what exactly is a perfect square trinomial? It is an expression that can be written in the form

• \(a^2 - 2ab + b^2\)
• or \(a^2 + 2ab + b^2\)

Think of it like a formula. In our given problem, the first three terms were \(x^2 - 2x + 1\). This matches perfectly to the structure of \(a^2 - 2ab + b^2\), where:

• \(a = x\)
• \(b = 1\)

Thus, it factors into \((x - 1)^2\), because squaring \((x - 1)\) gives you back the original expression. Recognizing these patterns is key to simplifying algebraic expressions and solving problems efficiently.

Difference of Squares

The difference of squares is another important tool when it comes to factoring. It's called a difference because it involves subtraction. Mathematically, it looks like \(a^2 - b^2 = (a - b)(a + b)\). This is a simplification tool that works because both a positive and a negative multiplication is involved.In the example problem, after recognizing the perfect square trinomial, the expression became \((x - 1)^2 - 9z^2\), which shows a difference of squares:

• \(a = x - 1\)
• \(b = 3z\)

This allows us to rewrite it as \((x - 1 - 3z)(x - 1 + 3z)\). Using this property helps us break down complex polynomials into more manageable expressions. Always look for the difference of squares when factoring, as it unveils simpler forms that are easier to work with.

Algebraic Expressions

Algebraic expressions form the backbone of algebra. They are a mix of numbers, variables, and operations. These expressions can be simplified or factored to reveal patterns or solutions to problems.In our case, the expression was \(x^2 - 2x + 1 - 9z^2\). Mindfully approached, we broke it down into recognizable patterns like

• perfect square trinomials
• difference of squares

Working with algebraic expressions involves understanding these fundamental concepts to facilitate their manipulation.Recognizing and transforming algebraic expressions into simpler forms paves the way for easier handling and solution of more complex equations. Always start by identifying familiar patterns in your problem, as it will clarify your approach in tackling algebraic challenges.