Problem 45 - Algebra A Combined Function

Question

Factor each completely. $$ x^{4}-81 $$

Step-by-Step Solution

Answer

\((x - 3)(x + 3)(x^2 + 9)\)

Step 1: Recognize the Form

The expression is a difference of squares: \(x^4 - 81\). Recall that the difference of squares formula is given by \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a^2 = x^4\) and \(b^2 = 81\).

Step 2: Apply the Difference of Squares Formula

Apply the difference of squares formula: \(x^4 - 81 = (x^2)^2 - 9^2 = (x^2 - 9)(x^2 + 9)\).

Step 3: Factor Further if Possible

Now, examine \(x^2 - 9\) which can also be factored further as it is also a difference of squares: \(x^2 - 9 = (x - 3)(x + 3)\). The other factor, \(x^2 + 9\), cannot be factored further using real numbers.

Step 4: Write the Complete Factored Form

Combine the results from the previous steps. The completely factored form of the original expression is \((x - 3)(x + 3)(x^2 + 9)\).

Key Concepts

Difference of Squares

The difference of squares is a specific algebraic pattern used in polynomial factoring. This formula is a handy trick for factoring expressions where two squared terms are subtracting one another. The formula is expressed as \(a^2 - b^2 = (a - b)(a + b)\). Essentially, it allows you to break down complex expressions into simpler binomials.
For example, if you have an expression like \(x^4 - 81\), you can recognize it as a difference of squares:

• \(a^2 = (x^2)^2\) which is \(x^4\)
• \(b^2 = 9^2\) which is \(81\)

Applying the formula, these are rewritten as \((x^2)^2 - 9^2\) and factored to \((x^2 - 9)(x^2 + 9)\). This simple yet powerful method helps in reducing the complexity of higher-degree polynomials for further analysis or solutions.

Polynomial Factoring

Polynomial factoring is the process of expressing a polynomial as the product of its factors. It simplifies polynomials into multiple, simpler expressions that, when multiplied, return the original polynomial. This is a foundational skill in algebra, aiding in solving polynomial equations and understanding their properties.
When we are factoring polynomials, such as \(x^4 - 81\), we aim to express this polynomial as products of its factors as simply as possible. The first step using the difference of squares, \((x^2 - 9)(x^2 + 9)\), already made great progress in breaking it down.
Next, recognizing further opportunities for factoring such as \(x^2 - 9 = (x - 3)(x + 3)\) simplifies it further. While sometimes possible for other polynomials or more advanced applications, \(x^2 + 9\) doesn’t separate into real number factors.
Factoring polynomials allows you to solve equations, graph functions, or simplify expressions. Each factor represents a solution or root of the polynomial, contributing to the overall understanding of an equation's behavior.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operational symbols. These expressions can represent real-world quantities in a compact form, and manipulating them is a key component of algebra.
In tackling algebraic expressions such as \(x^4 - 81\), identifying patterns like the difference of squares is integral for simplifying them. Making these expressions simple often involves factorization techniques that break them into manageable pieces.
Additional simplifications depend on recognizing compound forms and seeing how algebraic rules apply. For example, expressions like \(x^4\) require an understanding of bases and exponents, whereas constants like \(81\) simply reference fixed values. Recognizing patterns like perfect squares or cubes, differences or sums, plays a significant role in effective algebra manipulation.
Manipulating algebraic expressions by recognizing these patterns and using correct rules gives a powerful way to solve problems, whether they involve finding values, optimizing solutions, or modeling scenarios.