Exercises 107–109 will help you prepare for the material covered in the next section. Factor: x³+3x²−x−3

Verified step by step guidance

1. Group the terms in pairs to make factoring easier: (x^3 + 3x^2) + (-x - 3).
2. Factor out the greatest common factor (GCF) from each group: x^2(x + 3) - 1(x + 3).
3. Notice that (x + 3) is a common binomial factor in both terms.
4. Factor out the common binomial (x + 3): (x + 3)(x^2 - 1).
5. Recognize that x^2 - 1 is a difference of squares and factor it further: (x + 3)(x - 1)(x + 1).

Key Concepts

Polynomial Factoring

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This process helps simplify expressions and solve equations. Common methods include factoring by grouping, using special formulas, or identifying common factors.

Factoring by Grouping

Factoring by grouping is a technique used when a polynomial has four or more terms. The polynomial is split into groups, each group is factored separately, and then a common binomial factor is factored out. This method is useful for cubic polynomials like the given expression.

Identifying Common Factors and Patterns

Recognizing common factors or patterns such as the difference of squares, sum/difference of cubes, or factoring out the greatest common factor is essential. For the polynomial x^3 + 3x^2 − x − 3, grouping terms to find common binomial factors is key to factoring it completely.