In Exercises 95–104, factor completely.8x⁴ − x/8

Verified step by step guidance

1. Step 1: Identify the common factor in the expression. In this case, the common factor is \( x \).
2. Step 2: Factor out the common factor \( x \) from the expression \( 8x^4 - \frac{x}{8} \). This gives us \( x(8x^3 - \frac{1}{8}) \).
3. Step 3: Notice that the expression inside the parentheses, \( 8x^3 - \frac{1}{8} \), is a difference of cubes.
4. Step 4: Recall the formula for factoring a difference of cubes: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \).
5. Step 5: Apply the difference of cubes formula to \( 8x^3 - \frac{1}{8} \), where \( a = 2x \) and \( b = \frac{1}{2} \), to factor it completely.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial expression as a product of its factors. This process is essential for simplifying expressions and solving equations. Common techniques include factoring out the greatest common factor (GCF), using special products like the difference of squares, and applying the quadratic formula for higher-degree polynomials.

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that divides two or more numbers or terms without leaving a remainder. Identifying the GCF is a crucial first step in factoring, as it allows for simplification of the polynomial by removing common terms. For example, in the expression 8x⁴ - x/8, the GCF can be determined to facilitate further factoring.

Polynomial Degree and Terms

The degree of a polynomial is the highest power of the variable in the expression. Understanding the degree helps in determining the behavior of the polynomial and the methods used for factoring. In the expression 8x⁴ - x/8, recognizing that the highest degree is 4 indicates that it is a quartic polynomial, which may require specific techniques for complete factorization.