Problem 60 - Elementary Algebra

Question

Factor completely. $$ 81 x 8-1 $$

Step-by-Step Solution

Answer

The complete factorization is \((3x^4 - 1)(3x^4 + 1)\).

Step 1: Identify the Expression Type

The expression given is \(81x^8 - 1\). This looks like a difference of squares because it is of the form \(a^2 - b^2\), where \(a = (3x^4)\) and \(b = 1\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Applying this to our expression, we get:\[(3x^4)^2 - 1^2 = (3x^4 - 1)(3x^4 + 1)\]

Step 3: Check for Further Factorization

Each factor from the previous step, \(3x^4 - 1\) and \(3x^4 + 1\), should be checked for further factorization. Neither of these can be factored further over the integers, as they do not fit the pattern of simple formulas like the difference of squares again.

Key Concepts

Difference of Squares

The difference of squares is a handy formula in algebra that simplifies the factoring of certain types of expressions.
The general form is given by:

• The formula: \(a^2 - b^2 = (a-b)(a+b)\)
• This formula states that the difference between two perfect squares can be expressed as the product of a sum and a difference.
• It applies when both terms are perfect squares and the operation between them is subtraction.

In our exercise, the expression \(81x^8-1\) is recognized as a difference of squares.
Here the term \(81x^8\) can be seen as \((3x^4)^2\), and 1 as \(1^2\).
Applying the formula, the expression becomes \((3x^4 - 1)(3x^4 + 1)\).
This not only highlights the simplicity of using the difference of squares formula but also how powerful it can be when dealing with higher power polynomials.

Integer Factorization

Integer factorization refers to expressing a number or algebraic expression as a product of its factors.
These factors are most frequently integers.
Here are some essential aspects:

• Integer factorization is fundamental in algebra to simplify or solve expressions and equations.
• In the case of the expression \(81x^8 - 1\), the formula \((3x^4 - 1)(3x^4 + 1)\) uses integer components as it's expressed with whole numbers and variables combined in a structured form.
• After applying the difference of squares, further integer factorization is checked for but not always possible unless the resulting terms fit straightforward factorizable patterns.

Although the expression \(3x^4 - 1\) and \(3x^4 + 1\) cannot be broken down into simpler integer factors using the difference of squares or other basic formulas, reviewing the process and confirming it ensures mathematical accuracy.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators that represent a value or range of values.
Here's what you need to know:

• An algebraic expression can take many forms, including monomials, binomials, and polynomials, as well as their products and sums.
• Our expression, \(81x^8 - 1\), is considered a binomial polynomial since it consists of two terms.
• The understanding of binomials and their factorization into elementary components, such as through the difference of squares, is vital in algebra.

Utilizing this understanding helps simplify algebraic expressions, making them easier to solve.
This exercise effectively shows how the rules of algebra can be employed to transform a complex expression into a simpler product of terms, showcasing the importance of recognizing expression types.