Problem 109 - Elementary Algebra

Question

Factor out the GCF first and then factor by grouping. $$ 24 a 3 b 2-60 a 3 b+40 a b 2-100 a b $$

Step-by-Step Solution

Answer

Factor as \(4ab((3a^2 + 5)(2b - 5))\).

Step 1: Identify the Greatest Common Factor (GCF)

First, identify the GCF of all the terms in the expression. Looking at \(24a^3b^2\), \(60a^3b\), \(40ab^2\), and \(100ab\), the GCF is \(4ab\) because it is the highest factor common to all the terms.

Step 2: Factor Out the GCF

Once the GCF is identified, we factor it out of each term in the expression. The expression is modified as follows:\(24a^3b^2 - 60a^3b + 40ab^2 - 100ab = 4ab(6a^2b - 15a^2 + 10b - 25)\).

Step 3: Factor by Grouping

Now, focus on the expression inside the parenthesis: \(6a^2b - 15a^2 + 10b - 25\). We group the terms into two pairs:Group 1: \(6a^2b - 15a^2\) Group 2: \(10b - 25\) .

Step 4: Factor Each Group

Factor out the GCF from each grouped pair:From Group 1: \(6a^2b - 15a^2\), the GCF is \(3a^2\), so it becomes \(3a^2(2b - 5)\).From Group 2: \(10b - 25\), the GCF is \(5\), which makes it \(5(2b - 5)\).

Step 5: Combine the Grouped Factors

Notice that both grouped terms (\(3a^2(2b-5)\) and \(5(2b-5)\)) contain the common binomial factor \((2b - 5)\). Combine these to write it as:\(4ab[(3a^2 + 5)(2b - 5)]\).

Key Concepts

Factoring

Factoring is the mathematical process of breaking down an expression into simpler components, called factors, that when multiplied together produce the original expression. It is like finding the ingredients of a recipe. The process of factoring is crucial because it can simplify expressions and solve equations more easily.
Practically, when you factor an algebraic expression, you are looking for numbers, variables, or combinations that multiply together to recreate the original term.

• Simplifies expressions by breaking them down into products of simpler expressions.
• Helps in resolving many algebraic problems, making equations easier to solve.
• Common factoring techniques include taking out the greatest common factor and using methods like grouping.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number or expression that divides two or more numbers or terms without a remainder. It is like finding the biggest building block common to all numbers involved.
Finding the GCF is a preliminary step in many factoring problems and is essential in simplifying expressions. To find the GCF:

• List the factors or divisors of each term.
• Identify the common factors or divisors.
• Choose the largest factor that is present in all terms.

In the example, the GCF of the terms in the expression \(24a^3b^2\), \(60a^3b\), \(40ab^2\), and \(100ab\) was determined to be \(4ab\). This is the factor that can be taken out of each term, simplifying the given polynomial.

Grouping Method

The grouping method is a systematic way to factor polynomials, especially helpful when dealing with four or more terms. It involves rearranging and grouping terms in a way that each group can have a common factor.
For instance, suppose we have a polynomial expression that looks unwieldy. We can try factoring by grouping:

• Identify pairs of terms that might have common factors.
• Factor out the greatest common factor from each pair.
• Look for a common factor in the newly formed expression.

In the given example, after factoring out the GCF, the polynomial \(6a^2b - 15a^2 + 10b - 25\) was reorganized into groups \(6a^2b - 15a^2\) and \(10b - 25\). Each group had a GCF, and upon factoring these, a common binomial \(2b - 5\) emerged, allowing further simplification.

Polynomials

Polynomials are expressions made up of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. They are like the building blocks in algebra. Understanding polynomials is fundamental in algebra and calculus.
Polynomials can take many forms, often categorized by degree (the highest power of the variable). A polynomial's structure determines the methods used to factor it. Common forms of polynomials include linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on.

• The process of factoring a polynomial can involve identifying the greatest common factor, recognizing patterns, or using methods such as the grouping.
• Knowing how to factor polynomials is important for solving polynomial equations, analyzing graphs, and even in calculus for finding derivatives.

The expression in the exercise is an example of a polynomial that was effectively factored by identifying its GCF first, then using grouping.