Problem 59 - Elementary Algebra

Question

Factor completely. $$ x 8-1 $$

Step-by-Step Solution

Answer

\( x^8 - 1 = (x - 1)(x + 1)(x^2 + 1)(x^4 + 1) \).

Step 1: Check the expression

Review the given expression. It seems to be incorrectly formatted as there is an error in its notation. The correct expression is likely either \( x^8 - 1 \) or if simplified using numbers, perhaps \( 8x - 1 \). For the purpose of this explanation, I'll assume it's meant to factor \( x^8 - 1 \).

Step 2: Recognize the difference of squares

Notice that the expression \( x^8 - 1 \) is a difference of squares. This can be expressed as \( (x^4)^2 - 1^2 \), which can be factored using the difference of squares formula \( a^2 - b^2 = (a-b)(a+b) \).

Step 3: Apply the difference of squares formula

Factor \( x^8 - 1 \) as \( (x^4 - 1)(x^4 + 1) \) using the difference of squares formula, where \( a = x^4 \) and \( b = 1 \).

Step 4: Factor \( x^4 - 1 \) using difference of squares

The expression \( x^4 - 1 \) itself can be further factored as it is also a difference of squares: \( (x^2)^2 - 1^2 = (x^2 - 1)(x^2 + 1) \).

Step 5: Factor \( x^2 - 1 \) using difference of squares

Further factor \( x^2 - 1 \) as \( (x - 1)(x + 1) \), which is a simple difference of squares where \( a = x \) and \( b = 1 \).

Step 6: Express the complete factorization

The complete factorization of \( x^8 - 1 \) is \( (x - 1)(x + 1)(x^2 + 1)(x^4 + 1) \). Note that \( x^2 + 1 \) and \( x^4 + 1 \) cannot be factored further over the real numbers.

Key Concepts

Factoring

Factoring is a core concept in algebra that involves rewriting an expression as a product of its factors. It is a crucial skill because it allows for simplification and can make solving equations easier. When given a polynomial, you look for common factors, differences of squares, or specific patterns that can be factored. For instance:

• Identify common factors: When numbers or terms are common across all parts of the expression, they can be taken out as a factor.
• Look for patterns: The difference of squares, perfect square trinomials, or other recognizable patterns help us factor more complex expressions easily.
• Break down the expression: Sometimes, a polynomial might need to be broken down into simpler parts before recognizing how to factor it.

Factoring transforms expressions by breaking them down into the smaller pieces that, when multiplied together, give back the original expression.

Difference of Squares

The difference of squares is a specific factoring pattern where a polynomial is expressed in the form of two squared terms separated by a minus sign. It is important to recognize because it simplifies the factoring process significantly. The formula for factoring a difference of squares is given by:\[ a^2 - b^2 = (a - b)(a + b) \].
For example, in the expression \( x^8 - 1 \), this can initially be seen as a difference of squares \((x^4)^2 - 1^2\).
Applying the formula, it breaks down into two factors: \( (x^4 - 1)(x^4 + 1) \).
The expression \( x^4 - 1 \) is again a difference of squares, allowing us to further factor it into \( (x^2 - 1)(x^2 + 1) \), with \( x^2 - 1 \) breaking down to \( (x - 1)(x + 1) \).Understanding the difference of squares not only helps in simplifying polynomials but also in solving various algebraic equations, making it an essential tool in a mathematician's toolkit.

Polynomials

Polynomials are algebraic expressions that consist of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. Understanding polynomials is fundamental in algebra as they appear frequently in many mathematical problems.A polynomial can take many forms and degrees:

• A monomial, such as \(3x^2\), is a polynomial with one term.
• A binomial, like \(x^2 - 1\), has two terms.
• A trinomial, such as \(x^2 + 5x + 6\), includes three terms.

In the original exercise, we worked with the polynomial \(x^8 - 1\), which represents a binomial and requires factoring to simplify.
Key points about polynomials include identifying highest exponents (degree), arranging terms in standard form (highest to lowest exponents), and understanding operations like addition, subtraction, and factoring.
With practice, breaking down polynomials through factoring or simplifying expressions becomes second nature, aiding in solving complex algebraic problems efficiently.