In Exercises 11–16, factor by grouping. x³−3x²+4x−12

Verified step by step guidance

1. Group the terms into two pairs: \( (x^3 - 3x^2) + (4x - 12) \).
2. Factor out the greatest common factor (GCF) from each group: \( x^2(x - 3) + 4(x - 3) \).
3. Notice that \( (x - 3) \) is a common factor in both terms. Factor \( (x - 3) \) out: \( (x - 3)(x^2 + 4) \).
4. Verify the factorization by distributing \( (x - 3) \) back into \( (x^2 + 4) \) to ensure it equals the original expression.
5. The expression is now factored as \( (x - 3)(x^2 + 4) \). This is the final factored form.

Key Concepts

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials by rearranging and grouping terms in pairs. This technique involves identifying common factors in each group, allowing for the extraction of a common binomial factor. It is particularly useful for polynomials with four or more terms, as it simplifies the expression into a product of simpler polynomials.

Common Factors

A common factor is a number or variable that divides two or more terms without leaving a remainder. In the context of polynomials, identifying common factors within grouped terms is essential for simplifying the expression. Recognizing these factors allows for the extraction of a binomial or monomial, which is a crucial step in the factoring process.

Polynomial Expressions

Polynomial expressions are mathematical expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Understanding the structure of polynomials, including their degree and coefficients, is vital for applying factoring techniques. In the given expression, recognizing the degree and the arrangement of terms aids in effectively applying the grouping method.