Problem 2 - Algebra for College Students

Question

Perform the indicated divisions of polynomials by monomials. $$ \frac{12 x^{3}-24 x^{2}}{6 x^{2}} $$

Step-by-Step Solution

Answer

The solution is \(2x - 4\).

Step 1: Identify the Division Expression

The expression given for division is \( \frac{12x^3 - 24x^2}{6x^2} \). This means we need to divide each term in the numerator by the monomial in the denominator.

Step 2: Separate the Terms

Write the expression as separate divisions for each term in the numerator:\[\frac{12x^3}{6x^2} - \frac{24x^2}{6x^2}\]

Step 3: Simplify Each Term Individually

Simplify each fraction individually:1. \( \frac{12x^3}{6x^2} = 2x \)2. \( \frac{24x^2}{6x^2} = 4 \)

Step 4: Combine the Simplified Terms

Combine the simplified terms from the previous step:\[2x - 4\]

Key Concepts

Understanding Monomials

A **monomial** is a type of algebraic expression that consists of only one term. It is formed by multiplying numbers and variables together. For example, the expression \(6x^2\) is a monomial. **Key characteristics of monomials include:**

• They contain only one term.
• They do not include any addition or subtraction operators.
• They can be a constant number, a variable, or a product of numbers and variables.

In the given exercise, the denominator \(6x^2\) is a monomial. Recognizing monomials is important in algebra to simplify expressions and solve equations. Each part of a monomial, like the numerical coefficient or the variables, can be manipulated in algebraic operations, such as division.

Simplifying Fractions

In algebra, **simplifying fractions** often involves breaking down complex expressions into simpler forms, especially when dividing polynomials by monomials. When simplifying the fraction \(\frac{12x^3}{6x^2}\), we consider both the numerical and variable parts separately.

**Steps to simplify:**

• Divide the coefficients (numerical values). For \(12x^3\), divide 12 by 6, resulting in 2.
• Examine the variable part. For \(x^3\) divided by \(x^2\), subtract the exponents: 3 - 2 = 1, which leaves us with \(x\).
• Combine the results to get \(2x\).

The key to simplifying fractions like \(\frac{24x^2}{6x^2}\) is realizing that dividing identical powers of a variable cancels them out.By understanding and practicing simplifying fractions with monomials, you will find it much easier to work through various algebra problems.

Working with Algebraic Expressions

**Algebraic expressions** are mathematical phrases that can include numbers, variables, and operators like addition and subtraction. In our task, the algebraic expression \(12x^3 - 24x^2\) is divided by the monomial \(6x^2\). **The Division of Algebraic Expressions Involves:**

• Breaking down complex expressions into simpler components, much like dividing fractions.
• Separating terms that can be individually simplified. This approach was exemplified in the exercise when transforming the division to involve each separate term as its division.
• Combining terms after simplification, such as combining \(2x\) and \(-4\), to achieve the final, simplified expression.

Algebraic expressions are foundational in solving equations and inequalities. Moving fluently between expressions and their simplified forms trains you to approach more challenging algebraic tasks with confidence.