Problem 38 - Algebra A Combined Function

Question

Factor each completely. $$ (x-1)^{2}-z^{2} $$

Step-by-Step Solution

Answer

\((x-1-z)(x-1+z)\)

Step 1: Recognize the Form

The expression \((x-1)^2 - z^2\) is a difference of squares. A difference of squares is a specific algebraic identity, which can be written as \(a^2 - b^2\). In our case, \(a = x-1\) and \(b = z\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a-b)(a+b)\). Substitute \(a = x-1\) and \(b = z\) into this formula. So, \((x-1)^2 - z^2 = ((x-1)-z)((x-1)+z)\).

Step 3: Simplify the Expression

Now simplify the equation from Step 2: Substitute back into the expression: \((x-1)^2 - z^2 = (x-1-z)(x-1+z)\). Ensure that there are no additional factors to consider and the expression can't be simplified further.

Key Concepts

Factoring

Factoring is a vital concept in algebra that involves breaking down expressions into simpler, multiplied components. It enables us to simplify expressions and solve equations more easily. When we talk about factored forms, we're describing how an expression like

• \(x^2 + 5x + 6\)

can be written as

• \((x+2)(x+3)\)

The purpose of factoring is to make complex expressions easier to handle and to solve particular mathematical problems.
In the context of our original exercise, the expression

• \((x-1)^2 - z^2\)

uses the difference of squares method, which aids in breaking the expression into two simpler binomials. Factoring the given expression allows us to work with it more easily, especially if further algebraic operations are needed.

Algebraic Identity

An algebraic identity is an equation that is true for all values of the variables involved. These identities help simplify mathematical expressions and provide tools for factoring.
One classic algebraic identity is the difference of squares formula:

• \(a^2 - b^2 = (a-b)(a+b)\)

In this formula, \(a^2\) represents the square of a number a, while \(b^2\) is the square of number b. The expression \(a^2 - b^2\) signifies a difference of two squares, hence, the name "difference of squares."
Understanding algebraic identities like this gives us a powerful tool for simplification and factoring. In our exercise, recognizing that

• \((x-1)^2 - z^2\)

is a difference of squares allows us to apply this identity directly, making algebraic manipulation much more straightforward.

Simplification

Simplification in algebra revolves around making expressions more manageable by reducing their complexity. We take bigger algebraic expressions and break them down into their simplest or most organized form.
One way to simplify is by identifying and utilizing patterns, like the difference of squares or other algebraic identities. In our exercise, we used this method to transform the expression

• \((x-1)^2 - z^2\)

into

• \((x-1-z)(x-1+z)\).

Simplification might also include combining like terms and making expressions easier to factor or solve. A well-simplified expression is not only easier to work with mathematically but also significantly reduces the possibility of errors during computations.

Expression Simplification

Expression simplification is more than just reducing the size or complexity of an expression; it's about making it as clear and concise as possible. This can involve factoring, using algebraic identities, and applying arithmetic rules to whittle down the terms.
In our given example, the essential process was recognizing the difference of squares and applying the formula to simplify the original expression

• \((x-1)^2 - z^2\)

to

• \((x-1-z)(x-1+z)\)

Once the expression is simplified, further operations, including solving equations or evaluating computations, become more straightforward. In homework or textbook exercises, mastering the skill of expression simplification can save time and reduce potential for mistakes.