In Exercises 69–78, factor each polynomial.x³ − 5 + 4x³y − 20y

Verified step by step guidance

1. Step 1: Identify common factors in the polynomial terms. Notice that the terms can be grouped as \(x^3 - 5\) and \(4x^3y - 20y\).
2. Step 2: Factor out the greatest common factor from each group. For the first group \(x^3 - 5\), there is no common factor other than 1. For the second group \(4x^3y - 20y\), the common factor is \(4y\).
3. Step 3: Rewrite the polynomial by factoring out the common factor from the second group: \(x^3 - 5 + 4y(x^3 - 5)\).
4. Step 4: Notice that \(x^3 - 5\) is a common factor in both terms of the polynomial.
5. Step 5: Factor out \(x^3 - 5\) from the entire expression: \((x^3 - 5)(1 + 4y)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. This process is essential for simplifying expressions, solving equations, and analyzing polynomial behavior. Common methods include factoring out the greatest common factor (GCF), using special products like the difference of squares, and applying techniques such as grouping.

Greatest Common Factor (GCF)

The greatest common factor (GCF) of a polynomial is the largest polynomial that divides all terms of the polynomial without leaving a remainder. Identifying the GCF is often the first step in factoring, as it allows for simplification of the polynomial. For example, in the polynomial x³ + 4x³y - 5 - 20y, the GCF can help isolate common terms for easier factoring.

Grouping Method

The grouping method is a technique used to factor polynomials with four or more terms by grouping terms into pairs or sets that share a common factor. This method simplifies the polynomial into a form that can be factored further. For instance, in the polynomial x³ + 4x³y - 5 - 20y, grouping the terms strategically can reveal common factors that facilitate the factoring process.