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Chapter 5: Problem 144

Factor each polynomial completely. $$x^{4}-1$$

Short Answer

Expert verified
\( x^{4} - 1 = (x - 1)(x + 1)(x^{2} + 1) \)

Step by step solution

01

Recognize a difference of squares

The given polynomial \[ x^{4} - 1 \] is a difference of squares because it can be written as \[ (x^{2})^2 - 1^2 \]. This fits the form \( a^2 - b^2 \), which can be factored as \( (a - b)(a + b) \).
02

Apply the difference of squares formula

Using the difference of squares formula on \[ x^{4} - 1 = (x^{2})^2 - 1^2 \], factor it as \[ (x^{2} - 1)(x^{2} + 1) \]
03

Factor the difference of squares again

Observe that \[ x^{2} - 1 \] is also a difference of squares because it can be written as \[ (x)^2 - (1)^2 \]. Thus, it can be factored further using the difference of squares formula as \[ (x - 1)(x + 1) \]
04

Combine all factors

Combine the factors from steps 2 and 3. The completely factored form of \[ x^{4} - 1 \] is \[ (x - 1)(x + 1)(x^{2} + 1) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

difference of squares
The 'difference of squares' is a fundamental concept in algebra, crucial for polynomial factorization. It is used when a polynomial is expressed as the difference between two squared terms. The general formula is \(a^2 - b^2 = (a - b)(a + b)\). For example, given \(x^4 - 1\), we rewrite it as \((x^2)^2 - 1^2\). This clearly shows the difference of two squares, where \(a = x^2\) and \(b = 1\). Applying the formula results in \((x^2 - 1)(x^2 + 1)\). This technique simplifies complex polynomials into more manageable factors.
polynomial factorization
Polynomial factorization involves rewriting a polynomial as a product of its simpler polynomials. This process is essential in solving polynomial equations and simplifying algebraic expressions. For the polynomial \(x^4 - 1\), after recognizing that it is a difference of squares, we applied the difference of squares formula: first, \((x^2 - 1)(x^2 + 1)\). Next, the term \(x^2 - 1\) itself is a difference of squares, and can be factored further as \((x - 1)(x + 1)\). The final factorization of \(x^4 - 1\) becomes \((x - 1)(x + 1)(x^2 + 1)\). Each step breaks down the polynomial into simpler, multiplied forms, making it easier to handle.
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations. Understanding how to manipulate these expressions is key to mastering algebra. For instance, in the expression \(x^4 - 1\), we apply multiple algebraic techniques for simplification. First, identify patterns like the 'difference of squares'. Then we factorize using known formulas. Each operation aims to transform the expression into a product of simpler factors. Breaking down \(x^4 - 1\) reveals the structure \((x - 1)(x + 1)(x^2 + 1)\). This approach not only simplifies complex problems but also helps in solving equations and understanding deeper algebraic concepts.

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