Problem 48 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ 4 n^{2}-64 $$

Step-by-Step Solution

Answer

The factored form of the quadratic expression \(4n^{2}-64\) is \((2n+8)(2n-8)\).

Step 1: Identify the Square Roots

Identify the square roots of each term in the expression. The square root of \(4n^{2}\) is \(2n\) and the square root of \(64\) is \(8\). So, the expression \(4n^{2}-64\) is equivalent to \((2n)^{2} - (8)^{2}\).

Step 2: Apply Difference of Squares Formula

Applying a difference of squares formula, \((a)^{2} - (b)^{2} = (a+b)(a-b)\), where, in this case, \(a\) is \(2n\) and \(b\) is \(8\). This leads to \((2n+8)(2n-8)\).

Step 3: Simplify the Expressions

Simplify the expressions in both brackets. The simplified expression becomes \((2n+8)(2n-8)\).

Key Concepts

Difference of Squares

When it comes to factoring, the difference of squares is a fundamental concept. It refers to an expression that involves two squared terms separated by a minus sign. The general formula is

• \(a^2 - b^2 = (a + b)(a - b)\)

Understanding this formula is crucial as it allows you to simplify complex expressions into easier factors.


In our exercise, the expression \(4n^2 - 64\) is a classic example. Here,

• \(4n^2\) can be expressed as \((2n)^2\)
• \(64\) can be expressed as \((8)^2\)

Thus, it perfectly fits the difference of squares framework, enabling easy factoring using the formula. This not only simplifies calculations but also helps in understanding the structure of algebraic expressions better.

Quadratic Expressions

A quadratic expression is a polynomial expression of degree 2, meaning the highest exponent of the variable is 2. These expressions typically take the form

• \(ax^2 + bx + c\)

where \(a\), \(b\), and \(c\) are constants.

In the given exercise, \(4n^2 - 64\) might not appear as a conventional quadratic expression due to its missing linear (\(bx\)) term. However, it still qualifies as quadratic since the highest power of \(n\) is 2.

Factoring quadratic expressions can solve equations or find roots more easily. Recognizing patterns such as the difference of squares speeds up this process significantly. By factoring, we've transformed the quadratic \(4n^2 - 64\) into the product form \((2n + 8)(2n - 8)\), simplifying the expression.

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients. These expressions are structured through addition, subtraction, and multiplication operations, often arranged in ascending order of degrees.

The provided expression, \(4n^2 - 64\), is classified as a polynomial. Although minimal in complexity, it still highlights the essential polynomial pattern.

• This particular polynomial features two terms: \(4n^2\) and \(-64\).
• The degree of the term \(n\) partially defines its categorization.

Transforming polynomials through factoring is a critical skill. It aids in not only simplifying but also analyzing the behavior and properties of higher degree expressions. In our case, factoring reveals that \(4n^2 - 64\) is a difference of two squares, hence its breakdown into linear factors \((2n + 8)(2n - 8)\). This exemplifies how polynomials can be reinterpreted through mathematical techniques.