Problem 52 - Elementary and Intermediate Algebra

Question

Factor. See Example 5 or Example 7. $$ -28+6 a^{2}-2 a $$

Step-by-Step Solution

Answer

The expression factors to \(2(3a - 7)(a + 2)\).

Step 1: Rearrange the Terms

Reorder the terms of the polynomial \[ -28+6a^2-2a \]to make it easier to identify a common factor:\[ 6a^2 - 2a - 28 \].

Step 2: Find the Greatest Common Factor (GCF)

Identify the greatest common factor of the terms:For \(6a^2\), \(-2a\), and \(-28\), the GCF is 2.Factor out the GCF:\[ 2(3a^2 - a - 14) \].

Step 3: Factor the Quadratic Expression

Focus on factoring the expression within the parentheses:\[ 3a^2 - a - 14 \].Look for two numbers that multiply to \(3 imes -14 = -42\) and add to \(-1\). These numbers are \(-7\) and \(6\), so rewrite the middle term:\[ 3a^2 - 7a + 6a - 14 \].

Step 4: Apply the Grouping Method

Use grouping to factor by grouping terms:- Group as \((3a^2 - 7a) + (6a - 14)\).- Factor each group: \( a(3a - 7) + 2(3a - 7) \).- The common factor in each group is \((3a - 7)\), hence factor it out:\[ (3a - 7)(a + 2) \].

Step 5: Combine the Factors

Combine all the factors:The expression now is:\[ 2(3a - 7)(a + 2) \].This is the fully factored expression of the original polynomial.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a key concept when factoring polynomials. It represents the largest factor shared by all terms in a polynomial expression. Let's see how this works with our example.For instance, consider the polynomial terms: - \(6a^2\) - \(-2a\) - \(-28\)The primary task is to determine the largest factor that divides each of these terms completely. Both \(a^2\) and \(-2a\) suggest that "\(2\)" is common, as it divides each term without leaving a remainder.To find the GCF:

• List the factors of each coefficient: \(6\) (factors: \(1, 2, 3, 6\)), \(-2\) (factors: \(1, 2\)), and \(-28\) (factors: \(1, 2, 4, 7, 14, 28\)).
• The greatest common factor among these numbers is \(2\).

Rewriting the original polynomial \(-28 + 6a^2 - 2a\) with the GCF factored out simplifies the problem. By extracting 2, the expression becomes \(2(3a^2 - a - 14)\), making the next steps more manageable.

Quadratic Expression

Quadratic expressions are polynomials of the second degree and take the general form \(ax^2 + bx + c\). Factoring such expressions is an essential algebraic skill, as it allows us to rewrite the polynomial as a product of factors.For the quadratic expression \(3a^2 - a - 14\), we need to find two numbers that multiply to the product of the coefficient of \(a^2\) (which is \(3\)) and the constant term \(-14\). This product is \(-42\).Next, we look for two numbers that multiply to \(-42\) and also add up to the middle term coefficient, \(-1\). The numbers that satisfy these conditions are \(-7\) and \(6\). These numbers help us break down the middle term:- Rewrite \(-a\) as \(-7a + 6a\).This changes our expression from \(3a^2 - a - 14\) into \(3a^2 - 7a + 6a - 14\). This step is crucial because it sets the stage for the Grouping method, allowing us to factor the quadratic expression more effectively.

Grouping Method

Grouping is a method used for factoring polynomials, especially when dealing with quadratic expressions that don't readily decompose into obvious factors. In our example, we were given \(3a^2 - 7a + 6a - 14\). The Grouping method facilitates breaking this expression into smaller parts that can be factored independently:1. **Form Groups:** - Divide the expression into two pairs: \( (3a^2 - 7a) + (6a - 14) \).2. **Factor Each Group:** - From \(3a^2 - 7a\), factor out \(a\): \(a(3a - 7)\). - From \(6a - 14\), factor out \(2\): \(2(3a - 7)\).3. **Find and Factor Out the Common Binomial:** - Both pairs share a common factor \((3a - 7)\). - Thus, factor \((3a - 7)\) out: \((3a - 7)(a + 2)\).Utilizing the Grouping method helps to break down complex expressions into products of simpler factors, an essential step in algebraic simplification. Finally, combine this result with the previously factored GCF, yielding the complete factored form \(2(3a - 7)(a + 2)\).