Problem 67 - College Algebra with Modeling and Visualization

Question

Factor the expression completely, if possible. \(16 x^{4}-y^{4}\)

Step-by-Step Solution

Answer

\((2x - y)(2x + y)(4x^2 + y^2)\)

Step 1: Identify the Type of Expression

The expression given is \(16x^4 - y^4\). It is a difference of squares, as it can be rewritten as \((4x^2)^2 - (y^2)^2\). Difference of squares can be factored using the formula \(a^2 - b^2 = (a-b)(a+b)\).

Step 2: Apply the Difference of Squares Formula

Apply the formula to \(16x^4 - y^4\) by identifying \(a = 4x^2\) and \(b = y^2\). Therefore, \(16x^4 - y^4 = (4x^2 - y^2)(4x^2 + y^2)\).

Step 3: Check for Further Factoring

Now, inspect each factor for further factoring. The expression \(4x^2 - y^2\) is also a difference of squares because \((2x)^2 - (y)^2\).

Step 4: Factor the Further Difference of Squares

Factor \(4x^2 - y^2\) using the formula for difference of squares: \((2x - y)(2x + y)\). The expression \(4x^2 + y^2\) is a sum of squares, which typically cannot be factored further with real numbers.

Step 5: Combine the Factors

Combine all parts; the fully factored form of the expression is \((2x - y)(2x + y)(4x^2 + y^2)\).

Key Concepts

Difference of Squares

One of the fascinating concepts in algebra is the **difference of squares**. It revolves around expressions that can be written in the form of \( a^2 - b^2 \). This is one of the most straightforward factoring cases because it produces a very neat result. The formula to factor a difference of squares is:

• \(a^2 - b^2 = (a - b)(a + b)\).

The expression you start with, like \(16x^4 - y^4\), first needs to be rewritten to fit this model. In this example:

• Identify \(a = (4x^2)\) and \(b = (y^2)\).
• The expression can be written as \((4x^2)^2 - (y^2)^2\).

By applying the difference of squares formula, the expression simplifies nicely into \((4x^2 - y^2)(4x^2 + y^2)\). The beauty of the difference of squares lies in its simplicity and power within algebra. It provides a foundation for more complicated expressions, making them easier to handle and comprehend.

Algebraic Expressions

In mathematics, **algebraic expressions** are combinations of variables, numbers, and operators, such as addition or subtraction. Consider them as the language of algebra. In the exercise, the expression is \(16x^4 - y^4\), crafted from:

• Coefficients: Professionally, numbers placed in front of variables, like the 16.
• Variables: Symbols like \(x\) and \(y\) that represent unspecified numbers.
• Exponents: Powers that indicate the number of times a variable is multiplied by itself, such as the 4 in \(x^4\).

To navigate algebra successfully, it’s necessary to decipher these expressions and recognize patterns like the difference of squares mentioned earlier.
For instance:

• Identify that the expression is composed of terms like \(16x^4\) and \(-y^4\).
• Understanding terms and their roles makes it simpler to apply different mathematical rules and techniques effectively.

Algebraic expressions are foundational to more advanced mathematics and practical problem-solving, so grasping such basic concepts early on is very beneficial.

Factoring Techniques

A crucial skill in algebra involves mastering various **factoring techniques**. These techniques help simplify complex algebraic expressions by breaking them down into more manageable parts.
Here's an overview focusing on the problem \(16x^4 - y^4\):First, apply the difference of squares technique:

• Recognize it as \((4x^2)^2 - (y^2)^2\).
• Factor it into \((4x^2 - y^2)(4x^2 + y^2)\).

Next, inspect if individual factors can be further factored, especially those that still look like difference of squares:

• For example, \(4x^2 - y^2\) is also a difference of squares.
• Factor it as \((2x - y)(2x + y)\).

However, note that \(4x^2 + y^2\) represents a sum of squares, which cannot be factored further over real numbers. This limitation emphasizes the importance of recognizing and applying the correct technique based on the structure of the algebraic expression.
Each factoring technique, when used correctly, simplifies equations and fosters a deeper understanding of mathematical relationships.