Problem 10 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Factor out the greatest common factor. $$ 4 u v^{2}(2 u-v)+6 u^{2} v(v-2 u) $$

Step-by-Step Solution

Answer

The short answer for the given expression is: \(2uv((2u - v)(3u^{2} v + 2 u v))\).

Step 1: Identify the terms and their factors

In the given expression, we have two terms: 1. \(4 u v^{2} (2u-v)\) 2. \(6 u^{2} v (v-2u)\) Now, let's find their factors. Factors of the first term: 4, u, \(v^{2}\), \((2u-v)\) Factors of the second term: 6, \(u^{2}\), v, \((v-2u)\)

Step 2: Find the greatest common factor

From the list of factors in step 1, we can see that the greatest common factor is \(2uv\), as it divides both terms in the given expression.

Step 3: Factor out the greatest common factor

Now, let's factor out the GCF, \(2uv\), from the given expression: \(2uv (2 u v(2u - v) + 3u^{2} v(v-2u))\)

Step 4: Simplify the expression inside the parenthesis

Inside the parenthesis, we have a common factor \((2u-v)\) that can be taken out: \(2uv((2u - v)(2 u v + 3u^{2} v))\) Optional: You may choose to re-order the terms of the factor inside parenthesis for the final expression. Thus, the factored form of the given expression is: \(2uv((2u - v)(3u^{2} v + 2 u v))\)

Key Concepts

Greatest Common Factor

Understanding the concept of the greatest common factor (GCF) is essential in algebra for simplifying expressions and solving equations. The GCF of two or more numbers is the largest number that divides each of the numbers without leaving a remainder.

For algebraic expressions, the GCF extends to include variables and their exponents. It is found by identifying the smallest exponent of common variables across terms. For instance, in the expression presented, with terms including variables like 'u' and 'v', you look for the lowest powers of these variables that appear in each term.

To extract the GCF from an algebraic expression effectively:

• First, list out the factors of each term, as shown in the step-by-step solution.
• Then, identify common factors across these terms.
• Lastly, choose the largest factor that is common to all, keeping the lowest powers of any variables.

This strategy simplifies the expression and prepares it for further algebraic manipulation, as it reduces the complexity of the terms involved.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They are fundamental in algebra to describe relationships and to formulate equations. An expression can consist of single terms (monomials), two terms (binomials), several terms (polynomials), and more complex forms.

In our exercise example, we deal with a polynomial that has two terms. Each term is itself a product of numerical coefficients and variables to various powers. Understanding how to manipulate these expressions is vital for solving equations and simplifying complex algebraic expressions.

When working with algebraic expressions, it is important to recognize structures like:

• The coefficients (numeric factors in front of variables).
• The variables (letters used to represent unknown values).
• The exponents (numbers that denote how many times a variable is multiplied by itself).

By identifying these components, you're better equipped to employ operations such as factoring, distribution, and combining like terms, to manage and simplify these expressions.

Simplifying Expressions

Simplifying expressions is a process used in algebra to reduce complexity and present expressions in the most basic form. This involves factoring, combining like terms, and canceling out elements when possible.

In the solved exercise, simplification occurs after factoring out the GCF. Here's how you might further break down that process:

• Check for any common factors inside the parentheses for possible reduction.
• Look for terms that can be combined, such as like terms or those that are additive inverses.
• Use algebraic identities, if applicable, to further simplify or rearrange terms.

Simplifying expressions is an iterative process that works towards making expressions easier to interpret or solve. In the original exercise, the simplification led to a clearer structure by revealing a common binomial factor that further showcased the relationship between the two terms in the expression. Remember, the goal of simplifying is not only to make an expression shorter but also to make it clearer and more understandable.