In Exercises 11–16, factor by grouping. x^3−3x^2+4x−12

Verified step by step guidance

1. Step 1: The first step in factoring by grouping is to group the terms in pairs. In this case, we can group the terms as follows: (x^3−3x^2) + (4x−12).
2. Step 2: Next, factor out the greatest common factor (GCF) from each group. The GCF of x^3−3x^2 is x^2, and the GCF of 4x−12 is 4. So, we get: x^2(x−3) + 4(x−3).
3. Step 3: Now, you can see that (x−3) is a common factor in both terms. So, we can factor out (x−3) from the expression.
4. Step 4: After factoring out (x−3), we get: (x−3)(x^2+4).
5. Step 5: Finally, check your work by distributing (x−3) back into (x^2+4). If you get the original expression, then your factoring is correct.

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 looking for a 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 us to simplify expressions and reveal underlying structures. In the context of grouping, recognizing the common factors in each group is essential for successful factorization.

Polynomial Degree

The degree of a polynomial is the highest power of the variable in the expression. Understanding the degree helps in determining the number of roots and the behavior of the polynomial function. In the given polynomial, the degree is three, indicating it is a cubic polynomial, which influences the methods used for factoring and solving.