Problem 72 - Elementary Algebra

Question

Factor. Either factor out the greatest common factor, factor by grouping, use the guess and check method, or use the \(a c\) method. $$ 15 a b+5 a c-6 b-2 c $$

Step-by-Step Solution

Answer

(5a - 2)(3b + c)

Step 1: Identify and group terms

Identify pairs of terms that can be grouped based on common factors. Here, group **15ab + 5ac** and **-6b - 2c**.

Step 2: Factor out the common factor in each group

In the first group (15ab + 5ac), factor out 5a: **5a(3b + c)**. In the second group (-6b - 2c), factor out -2: **-2(3b + c)**.

Step 3: Combine the factored expressions

Since both groups contain the common factor **(3b + c)**, the original expression can be rewritten as the product of two binomials: **(5a - 2)(3b + c)**.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a valuable tool in factoring polynomials.
The GCF is the largest factor that divides all terms in a given polynomial.
To find it, identify all the factors common to each term. For instance, with the terms **15ab** and **5ac**, notice the common factor is **5a**. Factoring out the GCF simplifies the polynomial, making further steps easier.
In our exercise:

• For **15ab + 5ac**, the GCF is **5a**.
• For **-6b - 2c**, the GCF is **-2**.

Hence, we factor out these GCFs from each group.

Factoring by Grouping

Factoring by grouping is a method that involves grouping terms in pairs, then factoring out the common factor from each pair.
This method is particularly useful when dealing with longer polynomials.
Here's a step-by-step breakdown:

• First, group terms that have common factors. In our example, group **15ab + 5ac** and **-6b - 2c**.
• Next, factor out the common factor from each group. For **15ab + 5ac**, we factor out **5a** to get **5a(3b + c)**. For **-6b - 2c**, we factor out **-2** to get **-2(3b + c)**.
• Both resulting groups now share the common binomial factor **(3b + c)**. Combine these factored forms to get the final factored expression: **(5a - 2)(3b + c)**.

This step-by-step grouping and factoring out process simplifies the polynomial significantly.

Binomials

A binomial is a polynomial with exactly two terms.
It forms a key part of factoring expressions.
In our exercise, after grouping and factoring, we encountered the binomials **(3b + c)**.
Once we identified **(3b + c)** as a common factor in both groups, the original polynomial was simplified to a product of binomials: **(5a - 2)(3b + c)**.
Binomials are a core element in many polynomial factoring problems, and understanding how to work with them simplifies the factoring process significantly.
Remember, the product of two binomials can often be expanded using the distributive property or recognized through patterns such as the difference of squares or perfect square trinomials.