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. First, group the terms in pairs to make factoring easier: \(\left(x^{3} + 3x^{2}\right) + \left(-x - 3\right)\).
2. Next, factor out the greatest common factor (GCF) from each group: from the first group, factor out \(x^{2}\) to get \(x^{2}(x + 3)\); from the second group, factor out \(-1\) to get \(-1(x + 3)\).
3. Now, notice that both groups contain the common binomial factor \((x + 3)\), so factor this out: \((x + 3)(x^{2} - 1)\).
4. Recognize that \(x^{2} - 1\) is a difference of squares, which can be factored further as \((x - 1)(x + 1)\).
5. Write the fully factored form as \((x + 3)(x - 1)(x + 1)\).

Key Concepts

Polynomial Factoring

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This process helps simplify expressions and solve polynomial equations by breaking them down into factors that are easier to work with.

Grouping Method

The grouping method is a factoring technique where terms in a polynomial are grouped in pairs or sets to find common factors. This method is especially useful for polynomials with four or more terms, allowing you to factor by extracting common binomial factors.

Rational Root Theorem

The Rational Root Theorem helps identify possible rational roots of a polynomial by considering factors of the constant term and leading coefficient. Testing these roots can simplify factoring by finding zeros that correspond to linear factors.