Problem 70 - 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. $$ 12 a^{2} b^{3} c^{2} r^{7} ; a^{2} c^{2} r^{7} $$

Step-by-Step Solution

Answer

Answer: The coefficient of the group of factors \(a^{2}c^{2}r^{7}\) in the given term is 12.

Step 1: Identify the given term and the group of factors.

The given term is: $$ 12a^{2}b^{3}c^{2}r^{7} $$And the given group of factors is: $$ a^{2}c^{2}r^{7} $$

Step 2: Divide the given term by the group of factors.

In order to find the coefficient of the given group of factors, we need to divide the given term by the group of factors. So we divide: $$ \frac{12a^{2}b^{3}c^{2}r^{7}}{a^{2}c^{2}r^{7}} $$

Step 3: Simplify the expression.

When dividing, we can cancel out the common factors which are \(a^{2}\), \(c^{2}\), and \(r^{7}\). Thus we are left with: $$ \frac{12b^{3}}{1} = 12b^{3} $$

Step 4: Identify the coefficient.

The coefficient of the group of factors is the remaining constant in front of the variable(s). In this case, the coefficient of \(a^{2}c^{2}r^{7}\) in the given term is $$ 12 $$Since \(12b^{3}\) is the result when dividing the given term by the group of factors, and the constant \(12\) is in front of the variable \(b^{3}\).

Key Concepts

Polynomial division

Polynomial division is similar to long division in numbers, but instead of numbers, you deal with algebraic expressions. The goal is to divide two polynomials and simplify the result. In this exercise, we divided the given term by the group of factors.

When dividing polynomials, here's what you do:

• Identify the terms you are dividing.
• Write them as a fraction, placing the divisor in the denominator.
• Simplify by cancelling out common factors in the numerator and the denominator.

This process helps us find the remaining expression or the coefficient, as required in such exercises.

Understanding polynomial division is essential because it plays a key role not just in algebraic expressions but also in understanding and solving equations.

Coefficient

A coefficient is a number placed in front of a variable within an algebraic term. It signifies how many times the term is multiplied. In our example, we were asked to find the coefficient after dividing the term by specific factors.

In simpler terms:

• If you have a term like \(3x\), the coefficient is \(3\).
• In more complex terms like \(12a^{2}b^{3}c^{2}r^{7}\), if the factors \(a^{2}c^{2}r^{7}\) are divided out, the coefficient left is \(12\).

The coefficient tells us the intensity or quantity of a certain term without altering the variable part. Knowing how to isolate and identify coefficients after simplification is important in solving algebra problems.

Factors

Factors of an algebraic term are the numbers or variables that, when multiplied together, form the term itself. They play a crucial part in breaking down expressions into simpler parts, much like deconstructing a puzzle.

If you consider the term \(12a^{2}b^{3}c^{2}r^{7}\), its factors include:

• Numeric factors, such as \(12\).
• Variable factors, such as \(a^{2}\), \(b^{3}\), \(c^{2}\), and \(r^{7}\).

Understanding factors is vital for dividing expressions and finding commonalities or simplifying polynomials. Being proficient in identifying and working with factors can make complex algebraic problems more manageable and easier to solve.

Algebraic expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. They represent a quantity and can be simplified or manipulated to solve mathematical problems.

In algebra, expressions can take various forms such as:

• Simple expressions like \(x + y\).
• More complex ones like \(12a^{2}b^{3}c^{2}r^{7}\).

These can be constructed from constants and variables and manipulated through operations like addition, subtraction, multiplication, and division.

Learning to work with algebraic expressions is foundational in understanding broader algebra concepts, including polynomial division, coefficients, and factors. Mastery of these expressions is essential for solving equations and crafting detailed mathematical solutions.