Problem 42 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. $$2 u(v-7)+(v-7)$$

Step-by-Step Solution

Answer

The short answer is: \((v-7)(2u+1)\).

Step 1: Identify the terms and their factors

The given expression is: $$2u(v-7)+(v-7)$$ We can see two terms: 1. \(2u(v-7)\) 2. \((v-7)\) Now, let's identify the factors of each term. 1. First term (\(2u(v-7)\)) factors: \(2\), \(u\), and \((v-7)\) 2. Second term (\(v-7\)) factors: \((v-7)\)

Step 2: Find the greatest common factor

Looking at the factors of each term, we can see that they have a common factor - \((v-7)\). Thus, \((v-7)\) is the greatest common factor.

Step 3: Factor out the greatest common factor

We will now factor out the greatest common factor \((v-7)\) from the given expression. $$2u(v-7)+(v-7) = (v-7)(2u+1)$$ So, the factored form of the given expression is: $$(v-7)(2u+1)$$

Step 4: Check the answer by expanding the factored form

To check our answer, we will expand the factored form to see if we arrive back at the original expression. Expanding \((v-7)(2u+1)\), we get: $$ (v-7)(2u+1) = 2u(v-7)+(v-7)$$ Since the expanded form matches the original expression, our factored form \((v-7)(2u+1)\) is correct.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial technique in algebra that helps to simplify expressions and solve equations more efficiently. When we talk about factoring polynomials, we generally mean expressing a polynomial as a product of its factors. This can often make it easier to understand and work with polynomial expressions. One common method is factoring out the greatest common factor (GCF), which is the largest expression that divides all terms in the polynomial.
For the example given:

• The expression is \(2u(v-7) + (v-7)\).
• Both terms share a common factor \((v-7)\).
• This means \((v-7)\) is the GCF.

By factoring out \((v-7)\), we simplify the expression to \((v-7)(2u+1)\). This method is very useful as it helps reduce the complexity of expressions and equations in mathematics.

Algebraic Expressions

Algebraic expressions, like those in our exercise, consist of variables, numbers, and operations combined together. These expressions do not have an equality sign as seen in equations but instead are value placeholders that can be manipulated or evaluated.
Key elements of an algebraic expression include:

• **Variables:** Represent unknowns or placeholders, often denoted by letters (like \(u\) and \(v\) in our discussion).
• **Constants:** Fixed values (such as numbers outside the parentheses).
• **Operations:** Include addition, subtraction, multiplication, and division.

In algebra, the purpose is often to simplify expressions or solve for variables. Identifying common factors, like \((v-7)\) in our example, helps make simplifying expressions more straightforward.

Mathematics Education

Understanding concepts such as factoring and working with algebraic expressions forms the foundation of mathematics education, particularly in algebra. These concepts equip students with problem-solving skills and logical reasoning abilities necessary for advanced mathematical thinking.
Learning through exercises such as this one allows students to grasp the importance of methods like factoring in simplifying expressions and tackling more complex problems.

• **Practice:** Regular practice helps internalize these concepts.
• **Conceptual Understanding:** Focusing on understanding the 'why' and 'how' of algebraic manipulations.
• **Application:** Applying these concepts to real-world scenarios or more challenging math problems enhances learning.

By mastering these essential algebraic skills, students build confidence in their ability to engage with mathematics critically and productively. This not only aids in academic success but also empowers lifelong learning and logical reasoning skills.