In Exercises 45–68, factor by grouping.x³− 3x² + 4x − 12

Verified step by step guidance

1. Group the terms in pairs: \((x^3 - 3x^2) + (4x - 12)\).
2. Factor out the greatest common factor from each pair: \(x^2(x - 3) + 4(x - 3)\).
3. Notice that \((x - 3)\) is a common factor in both groups.
4. Factor out the common factor \((x - 3)\): \((x - 3)(x^2 + 4)\).
5. Verify the factorization by expanding \((x - 3)(x^2 + 4)\) to ensure it equals the original expression.

Key Concepts

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four or more terms. This technique involves rearranging the terms into two groups, factoring out the common factors from each group, and then factoring out the common binomial factor. It is particularly useful when the polynomial does not have a straightforward factorization.

Common Factors

A common factor is a number or variable that divides two or more terms without leaving a remainder. Identifying common factors is crucial in factoring polynomials, as it allows for simplification of expressions. In the context of grouping, recognizing the common factors in each group helps in breaking down the polynomial into simpler components.

Polynomial Expressions

Polynomial expressions are mathematical expressions that consist of variables raised to non-negative integer powers and their coefficients. They can be classified by their degree, which is the highest power of the variable. Understanding the structure of polynomial expressions is essential for applying various factoring techniques, including grouping.