Problem 20 - 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. $$ (-5) a^{5} b^{5} c^{5}, b c $$

Step-by-Step Solution

Answer

Answer: The coefficient of the group of factors \(bc\) in the term \((-5) a^{5} b^{5} c^{5}\) is \((-5) a^{5} b^{4} c^{4}\).

Step 1: Identify the given term and group of factors

The given term is \((-5) a^{5} b^{5} c^{5}\) and the group of factors is \(bc\).

Step 2: Rewrite the term in the form of (group of factors)(coefficient)

Divide the given term by the group of factors \(bc\) to find the coefficient: $$ \frac{(-5) a^{5} b^{5} c^{5}}{bc} = (-5) a^{5} b^{4} c^{4} $$ The coefficient of the group of factors \(bc\) in the term \((-5) a^{5} b^{5} c^{5}\) is \((-5) a^{5} b^{4} c^{4}\).

Step 3: State the final answer

The coefficient of the group of factors \(bc\) in the term \((-5) a^{5} b^{5} c^{5}\) is \((-5) a^{5} b^{4} c^{4}\).

Key Concepts

Coefficients

In algebra, coefficients are essential components that multiply the variables in terms of a polynomial. They serve as numerical or constant factors in these expressions. For example, in the term \(-5a^{5}b^{5}c^{5}\), the coefficient is \(-5\), because it multiplies the entire term. However, when you're working with a group of factors, the coefficient can take on a more complex form.

When dividing a polynomial term by a specified group of factors, the coefficient is what remains after dividing out these factors. In the exercise, dividing \((-5)a^{5}b^{5}c^{5}\) by \(bc\) results in the coefficient being \((-5)a^{5}b^{4}c^{4}\). Hence, identifying the correct coefficient is crucial for breaking down and analyzing polynomial terms in algebra.

Factorization

Factorization in algebra involves breaking down a complex expression into a product of simpler factors. These factors, when multiplied together, yield the original expression. It's like reverse-engineering a polynomial to understand its building blocks.

To factorize a polynomial term, consider its factors, such as numbers, variables, and even sub-terms. The exercise asks us to factor out \(bc\) from the term \((-5) a^{5}b^{5}c^{5}\), revealing the other factor as \((-5) a^{5}b^{4}c^{4}\). This step effectively decomposes the term to simplify the analysis or solve algebra problems.

Factorization is widely used for solving equations, simplifying expressions, and even finding roots of polynomials. Understanding this concept allows students to deal with more complex algebraic problems with confidence.

Polynomial Terms

Polynomial terms are the building blocks of a polynomial expression, consisting of products of coefficients and variables raised to positive integer powers. For example, \((-5) a^{5}b^{5}c^{5}\) is a polynomial term. Understanding the structure of these terms is crucial for performing arithmetic operations and simplifying expressions.

Each polynomial term consists of:

• A coefficient, which is a constant value.
• Variables, which are letters representing numbers.
• Powers/exponents, indicating how many times the variable is multiplied by itself.

When solving problems like the exercise given, it’s important to identify and manipulate each of these components correctly to find solutions efficiently.

These terms can combine through addition or subtraction to form polynomials, which are fundamental structures in algebra, used extensively in equations and functions in advanced mathematics.