Question

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

Short answer

(x - 1)(3x + 2)

Step-by-step solution

Group terms

Group the terms into two pairs: o Group the first two terms: o Group the last two terms: o This gives us: (3x^2 - 3x) and (2x - 2)

Factor out the greatest common factor (GCF)

Factor out the greatest common factor from each pair of terms: o From the first group (3x^2 - 3x), the GCF is 3x:o From the second group (2x - 2), the GCF is 2:o This results in:o 3x(x - 1) + 2(x - 1)

Factor by grouping the common binomial

Since there is a common binomial (x - 1) in both terms:o Factor out (x - 1): o (x - 1)(3x + 2)

Key concepts

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is an essential concept in algebra that helps simplify expressions and solve equations. It is the largest factor that divides two or more numbers. Identifying and factoring out the GCF is a crucial step in many algebraic processes, especially in factoring polynomials.

When you have an expression like the one in the exercise, which is \(3x^2 - 3x + 2x - 2\), you start by grouping the terms to find what each pair has in common. For instance, in the pair \(3x^2 - 3x\), both terms share a factor of \(3x\). Similarly, in the pair \(2x - 2\), both terms share a factor of \(2\).

Here’s how you factor out the GCF:

• From \(3x^2 - 3x\), you factor out \(3x\), resulting in \(3x(x - 1)\).
• From \(2x - 2\), you factor out \(2\), resulting in \(2(x - 1)\).

Notice how factoring out the GCF simplifies the expression, making it easier to see common factors and further simplify or solve the equation.

Binomials

A binomial is a polynomial with exactly two terms. In our exercise, after grouping and factoring out the GCF, we get \((x - 1)\) and \(3x + 2\) as our binomials.

Binomials are often the building blocks for more complex polynomials. When factoring by grouping, a common binomial indicates further simplification. In the example, both groups \(3x(x - 1)\) and \(2(x - 1)\) share the binomial \((x - 1)\).

Understanding binomials and recognizing common binomials within an expression is crucial because it lets you streamline your work, collapse terms, and eventually solve equations more efficiently.
To recap the binomials in the problem:

• From the first group, we have \(3x(x - 1)\), featuring the binomial \((x - 1)\).
• From the second group, we have \(2(x - 1)\), again featuring the binomial \((x - 1)\).

These shared binomials are then factored out to simplify the overall expression.

Factoring Techniques

Factoring is a technique used to simplify polynomials by expressing them as products of simpler polynomials. One common method is factoring by grouping, which is particularly useful when dealing with four-term polynomials.

In our exercise, we use the factoring by grouping technique, which involves three steps:

• Grouping terms: Divide the polynomial into two pairs. For instance, \(3x^2 - 3x\) and \(2x - 2\).
• Factoring out the GCF: Identify and factor out the GCF from each pair. From \(3x^2 - 3x\), the GCF is \(3x\); from \(2x - 2\), the GCF is \(2\).
• Factoring by grouping: Look for a common binomial and factor it out. Here, \(x - 1\) is the common binomial between \(3x(x - 1)\) and \(2(x - 1)\). This simplifies our expression to \((x - 1)(3x + 2)\).

This method is efficient because it reduces complex polynomials into products of simpler factors, making further computation or solving easier. Practicing this technique can build strong algebra skills and aid in solving more advanced math problems.