Problem 72 - Elementary Algebra

Question

For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors. $$ 10 x(x+7)^{2} ; 10(x+7) $$

Step-by-Step Solution

Answer

Answer: The coefficient of the group of factors \(10(x+7)\) in the term \(10x(x+7)^{2}\) is \((x(x+7))\).

Step 1: Identify the given term and factors

The given term is \(10x(x+7)^{2}\) and the given factors are \(10(x+7)\).

Step 2: Express the given term in the form of the given factors

We want to express the term \(10x(x+7)^{2}\) in the form of \(10(x+7)\) and its coefficient. To do this, let's observe the given term and the given factors: $$ 10x(x+7)^{2} = 10x(x+7)(x+7). $$ Notice that the given factors \(10(x+7)\) appears in the term. We can factor this out and express the term in the desired form: $$ 10x(x+7)^{2} = 10(x+7) \cdot (x(x+7)). $$

Step 3: Identify the coefficient

Now that we have expressed the term in the form of the given factors, we can see that the coefficient of the given factors \(10(x+7)\) is \((x(x+7))\). So, the coefficient of the given group of factors is: $$ x(x+7). $$

Key Concepts

Algebraic Factors

Algebraic factors are elements that are multiplied together to get a given expression or polynomial. When identifying factors in algebraic expressions, we aim to express the expression as a product of its components.
For example, in the expression \( 10x(x+7)^2 \), the term can be factored into simpler components. These components, like \( 10(x+7) \), are what we refer to as algebraic factors. It's important to note that factors can be constants, variables, or other expressions entirely.
To successfully identify algebraic factors, look for:

• Common terms that can be factored out.
• Simplifications that lead to products of terms.
• Groupings within parentheses that suggest multiplication.

Polynomial Expressions

Polynomial expressions are mathematical statements involving variables and constants combined using addition, subtraction, multiplication, and non-negative integer exponents. A polynomial can range from a simple expression like \( x + 1 \) to more complex ones like \( 10x(x+7)^2 \).
Understanding polynomial expressions involves recognizing these key characteristics:

• Each part of a polynomial separated by a plus or minus sign is called a term.
• The degree of a polynomial is determined by the highest power of the variable in the expression.
• Polynomials can often be rearranged or modified to reveal their underlying factors.

Knowing these fundamentals helps in simplifying calculations and understanding the structure of algebraic problems.

Factoring Techniques

Factoring techniques are the methods used to break algebraic expressions into simpler factors, making them easier to work with. There are several strategies to consider, especially when dealing with complex expressions like \( 10x(x+7)^2 \).
Some effective factoring techniques include:

• Common Factor Extraction: Identify a common factor in all the terms and factor it out, simplifying the expression.
• Grouping: This involves rearranging and grouping terms to reveal common factors or recognizable patterns.
• Special Products: Recognize patterns like the difference of squares, perfect square trinomials, and the sum/difference of cubes for quick factoring.

By mastering these techniques, one can efficiently simplify polynomials, solve equations, and identify coefficients as needed, like in the exercise above where we extracted \( 10(x+7) \) as factors and identified its coefficient.