Problem 46 - Algebra A Combined Function

Question

Factor each completely. $$ x^{4}-256 $$

Step-by-Step Solution

Answer

\((x-4)(x+4)(x^2+16)\) is the complete factorization.

Step 1: Recognize the Difference of Squares

The given expression is \( x^4 - 256 \). Notice that 256 can be written as \( 16^2 \). Therefore, \( x^4 - 256 \) is a difference of squares, which can be expressed as \( (x^2)^2 - 16^2 \).

Step 2: Apply the Difference of Squares Formula

Use the formula for the difference of squares, \( a^2 - b^2 = (a-b)(a+b) \). Here, \( a = x^2 \) and \( b = 16 \). Substitute these into the formula to get \( (x^2 - 16)(x^2 + 16) \).

Step 3: Factor the First Term Again

Look at \( x^2 - 16 \), which is again a difference of squares: \( x^2 - 4^2 \). Apply the difference of squares formula again: \( x^2 - 16 = (x-4)(x+4) \).

Step 4: Identify if Further Factorization is Possible

Inspect \( x^2 + 16 \) to see if it can be factored further. Since it doesn't fit the difference of squares (it is a sum), and it doesn't factor further over the real numbers, it remains as is.

Step 5: Combine All Factors

Now combine all the factors obtained: \( x^4 - 256 = (x-4)(x+4)(x^2+16) \).

Key Concepts

Difference of Squares

The difference of squares is a specific type of algebraic expression that takes the general form \( a^2 - b^2 \). This concept is a cornerstone in the process of factoring polynomials. It hinges on the idea that an expression involving squares can be simplified into two binomial expressions. The formula used for this purpose is \( a^2 - b^2 = (a-b)(a+b) \).

In the context of the exercise, we're looking at \( x^4 - 256 \). The challenge is to recognize that 256 is equivalent to \( 16^2 \), which allows us to rewrite the original expression as \( (x^2)^2 - 16^2 \). Identifying this as a difference of squares is crucial for moving forward with the factorization process.

Factoring expressions as a difference of squares simplifies calculations and reveals the basic factors, making subsequent factorization steps more manageable. By applying the formula, \( (x^2 - 16)(x^2 + 16) \), we start to break the expression into simpler components that can be further analyzed or simplified.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving variables, numbers, and operation symbols. These expressions can be simple, involving just a single term, or complex, involving multiple terms, like in our example of \( x^4 - 256 \).

Understanding algebraic expressions is vital for performing operations such as addition, subtraction, and particularly factorization. Each part of the expression, whether a term or coefficient, plays a role in how the expression can be manipulated or broken down. In the exercise, we are tasked to simplify the expression further by recognizing it as a difference of squares.

Mastery of interpreting and working with algebraic expressions allows one to solve equations and understand deeper mathematical concepts. In our specific scenario, effectively transforming \( x^4 \) into \( (x^2)^2 \) and recognizing 256 as \( 16^2 \) are steps grounded in interpreting the structure of algebraic expressions.

Factorization Techniques

Factorization techniques are methods used to express a polynomial as a product of its factors. These techniques are essential for simplifying algebraic expressions and solving polynomial equations. One of the most efficient techniques is identifying patterns such as the difference of squares.

In our example, \( x^4 - 256 \), we've utilized the difference of squares formula first to break it into \((x^2 - 16)(x^2 + 16)\). But we don't stop there. A key part of mastering factorization is recognizing when parts of the expression can be factored further. This is evident in \( x^2 - 16 \), which is itself a difference of squares that can be factored into \((x-4)(x+4)\).

The factor \( x^2 + 16 \), however, doesn't simplify further under real numbers, illustrating the importance of understanding the limitations of factorization. Developing a solid grasp of these techniques empowers students to handle more complex polynomials with confidence and clarity.