Question

Factor by grouping. $$ 3 x^{2}+6 x-x-2 $$

Short answer

(x + 2)(3x - 1)

Step-by-step solution

Group the terms

Group the terms in pairs to make factorization easier. Combine the first two terms and the last two terms: (3x^2 + 6x) + (-x - 2)

Factor out the common factor from each group

Factor out the greatest common factor (GCF) from each pair: 3x(x + 2) - 1(x + 2)

Factor out the common binomial factor

Notice that (x + 2) is a common factor in both terms. Factor it out: (x + 2)(3x - 1)

Key concepts

Understanding Algebra

Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. These symbols represent numbers and values in equations and expressions. By understanding algebra, you can solve problems involving unknown numbers. The most common symbols in algebra are letters from the alphabet, which are called variables. When working with variables, you can create expressions and equations to find unknown values.

Introduction to Factoring

Factoring is a process used in algebra to simplify expressions or solve equations by breaking down a complex expression into simpler factors. In this context, a factor is a number or expression that divides another number or expression evenly. Factoring is essential because it helps us solve quadratic equations, find the roots of polynomials, and simplify algebraic expressions. There are different methods of factoring, and one of them is called 'factoring by grouping.' In this method, you group terms in such a way that allows you to factor out the greatest common factor (GCF) from each group.

Polynomials and Their Characteristics

Polynomials are algebraic expressions that consist of variables and coefficients. These expressions can have multiple terms added together. Each term in a polynomial is a product of a number, called a coefficient, and a variable raised to a nonnegative integer power. The degree of a polynomial is determined by the highest power of the variable in the expression. When factoring polynomials like the one in the given exercise, we often aim to break them down into simpler binomials or trinomials. For example, in the exercise, we factor the polynomial \(3x^2 + 6x - x -2\) by grouping terms together.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor that can divide two or more numbers or terms without leaving a remainder. Finding the GCF is crucial in the process of factoring because it allows us to simplify expressions and make calculations more manageable. In the exercise, by grouping and factoring each pair of terms, we identify the GCF in each group. For instance, in the pair \((3x^2 + 6x)\), the GCF is \(3x\), and in the pair \((-x - 2)\), the GCF is \(-1\). Factoring out the GCF helps us to further simplify the polynomial.