In Exercises 11–16, factor by grouping. x³+6x²−2x−12

Verified step by step guidance

1. Group the terms into two pairs: \( (x^3 + 6x^2) - (2x + 12) \). This is the first step in factoring by grouping.
2. Factor out the greatest common factor (GCF) from each group. From the first group \( x^3 + 6x^2 \), the GCF is \( x^2 \), so it becomes \( x^2(x + 6) \). From the second group \( -2x - 12 \), the GCF is \( -2 \), so it becomes \( -2(x + 6) \).
3. Rewrite the expression with the factored groups: \( x^2(x + 6) - 2(x + 6) \).
4. Notice that \( (x + 6) \) is a common factor in both terms. Factor \( (x + 6) \) out: \( (x + 6)(x^2 - 2) \).
5. The expression is now fully factored as \( (x + 6)(x^2 - 2) \). Check your work by expanding the factors to ensure it matches 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 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 Degree

The degree of a polynomial is the highest power of the variable in the expression. Understanding the degree is essential for determining the behavior of the polynomial and for applying appropriate factoring techniques. In the given polynomial, the degree is three, indicating it is a cubic polynomial, which influences the methods used for factoring.