Problem 31 - College Algebra with Modeling and Visualization

Question

Use grouping to factor the polynomial. \(a b-3 a+2 b-6\)

Step-by-Step Solution

Answer

The polynomial is factored as \((a+2)(b-3)\).

Step 1: Group the Terms

The given polynomial is \( ab - 3a + 2b - 6 \). First, group the terms into two pairs: \((ab - 3a)\) and \((2b - 6)\).

Step 2: Factor Out Common Factors from Each Group

In the first group, \(ab - 3a\), the common factor is \(a\). Factoring \(a\) out, we get \(a(b - 3)\). In the second group, \(2b - 6\), the common factor is \(2\). Factoring \(2\) out, we get \(2(b - 3)\).

Step 3: Factor by Grouping

Now that each group contains the common binomial factor \((b-3)\), factor \((b-3)\) out of the expression: \(a(b-3) + 2(b-3) = (a+2)(b-3)\).

Step 4: Write the Final Factored Form

The polynomial \(ab - 3a + 2b - 6\) can be factored as \((a+2)(b-3)\).

Key Concepts

Grouping Method for Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of simpler expressions, and one effective technique for this is the grouping method. In this method, we rearrange the terms of a polynomial into groups that make it easier to factor.
Let's explore how this works step by step.

• First, identify if the polynomial can be divided into groups – usually pairs work best.
• Look for groups that allow you to factor out a common element within each group.

For example, with the polynomial \[ ab - 3a + 2b - 6 \]we start by grouping the terms as \[(ab - 3a) + (2b - 6).\]This breaks down the original expression into two smaller parts.
Groupings not only reduce complexity but also highlight common factors more clearly in each subset of terms.

Understanding Common Factors

Common factors are essential in the factoring process. A common factor is a term that appears in multiple expressions within the polynomial and can be factored out to simplify the overall expression. Here's how you identify and use them:

• Within each group of terms, check if the same factor appears in each term.
• Factor this common term out of the group.

In the example from before, examining the first group \[ ab - 3a, \] you'll notice the common factor is \[ a. \] Similarly, in the second group \[ 2b - 6, \]2 is the common factor. Pulling these out, we reach\[ a(b - 3) \] and\[ 2(b - 3). \]By extracting these common factors, you simplify each expression, paving the way for further factoring.

Achieving the Factored Form

The goal of factoring is to express the polynomial as a product of simpler expressions known as the factored form. After grouping and factoring out common elements, search for a common binomial across your grouped expressions.

• In this example, both groups share \[ (b-3). \]
• Factor out this shared binomial, then rearrange the expression to reveal its prime factors.

Following our example, \[ a(b-3) + 2(b-3) \] indicates that \[ (b-3) \] is common. Factor out \[ (b-3), \] leading to \[ (a+2)(b-3). \]This new expression, \[(a+2)(b-3),\]represents the polynomial in its factored form, which is neater and often easier to interpret or solve further if needed.