Problem 53 - Algebra A Combined Function

Question

Factor each completely. $$ 4 x^{2}+4 x+1-z^{2} $$

Step-by-Step Solution

Answer

The expression is factored as \((2x + 1 - z)(2x + 1 + z)\).

Step 1: Recognize the Expression

The given expression is a difference of squares: \( 4x^2 + 4x + 1 - z^2 \). Notice that \( 4x^2 + 4x + 1 \) can be represented as a perfect square trinomial, and \( z^2 \) is a perfect square.

Step 2: Identify the Perfect Square Trinomial

Rewrite \( 4x^2 + 4x + 1 \) as \( (2x + 1)^2 \). Check by expanding \((2x + 1)^2 = 4x^2 + 4x + 1\). This confirms \( 4x^2 + 4x + 1 \) is a perfect square.

Step 3: Express as a Difference of Squares

Now the expression becomes \( (2x + 1)^2 - z^2 \). This is a classic difference of squares form \( a^2 - b^2 \), where \( a = 2x + 1 \) and \( b = z \).

Step 4: Apply Difference of Squares Formula

The difference of squares formula is \( a^2 - b^2 = (a - b)(a + b) \). Substitute \( a = 2x + 1 \) and \( b = z \) to get: \( (2x + 1 - z)(2x + 1 + z) \).

Step 5: Final Factored Expression

The completely factored form of the expression is \( (2x + 1 - z)(2x + 1 + z) \).

Key Concepts

Difference of Squares

The concept of difference of squares is commonly used in algebra to simplify expressions and solve equations. It typically involves expressions of the form \( a^2 - b^2 \).
This is one of the simplest and most straightforward factoring techniques available.To understand it better, let's consider the difference of squares formula:

• \( a^2 - b^2 = (a - b)(a + b) \)

This formula states that any expression that can be written as the difference between two perfect squares can be factored into two binomial expressions.
In our exercise, the term \( (2x + 1)^2 - z^2 \) is a difference of squares.
For example:

• If \( a = 2x + 1 \)
• And \( b = z \)

Then the factored form will be \( (a - b)(a + b) \), which simplifies to \( (2x + 1 - z)(2x + 1 + z) \).
Understanding this concept is crucial as it allows you to quickly break down complex algebraic expressions into simpler forms.

Perfect Square Trinomial

A perfect square trinomial is a special form of a trinomial that can be expressed as the square of a binomial. This occurs when trinomial expressions form a complete square, making them easier to simplify and factor.For example, let's take a look at the trinomial \( 4x^2 + 4x + 1 \). This expression is a perfect square trinomial because it can be rewritten as \( (2x + 1)^2 \).
Here’s why:

• The first term \( 4x^2 \) is the square of \( 2x \).
• The last term \( 1 \) is the square of \( 1 \).
• The middle term \( 4x \) is twice the product of these two terms, i.e., \( 2(2x)(1) = 4x \).

This pattern is consistent with the formula for a perfect square trinomial:

• \( (a + b)^2 = a^2 + 2ab + b^2 \)

Recognizing this pattern simplifies the process of factoring complex polynomials and helps in identifying underlying structures in expressions.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra and mathematics in general. They consist of numbers, variables, and operation symbols, all combined into a mathematical statement.An expression like \( 4x^2 + 4x + 1 - z^2 \) is a combination of different algebraic elements:

• \( 4x^2 \) and \( z^2 \) are terms involving the squares of variables.
• \( 4x \) is a linear term involving a single variable.
• \( 1 \) is a constant term, representing a fixed value.

Algebraic expressions can be manipulated using various algebraic techniques such as addition, subtraction, multiplying, expanding, and most importantly, factoring.
Factoring is particularly useful for breaking down complex expressions into simpler components. This can simplify solving equations and make understanding the relationships between different parts of the expression easier.Understanding the structure and components of algebraic expressions allows students to solve problems more efficiently and see the bigger picture in mathematical models.