Problem 23 - Technical Mathematics with Calculus

Question

$$3 a^{2} b+a b c-a b d$$

Step-by-Step Solution

Answer

3 a^{2} b + (c - d) ab

Step 1: Identify Like Terms

First, identify terms that have the same variables raised to the same powers. In this expression, the like terms are those with the variable 'ab' which are ab and a b d.

Step 2: Combine Like Terms

Combine the like terms by keeping the common variable 'ab' and summing the numerical coefficients. The terms a b c and -a b d combine to c d.

Step 3: Write the Simplified Expression

Keep the unchanged term and the result from the combination of like terms to write the simplified expression: 3 a^{2} b + (c - d) ab.

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operations (such as add, subtract, multiply, and division). These expressions represent quantities without a fixed value, often used in equations or formulas. For example, in the expression 3a^2b + abc - abd, we have variables a, b, c, and d combined with coefficients and exponents to form terms of the expression.

It's crucial to understand that the parts of these expressions which are 'like terms' can be combined. Like terms are terms whose variables (and their exponents) match exactly. Recognizing like terms allows us to simplify the expression to its shortest form, making it easier to work with in further calculations or applications.

Simplifying Expressions by Combining Like Terms

Simplifying expressions involves reducing them to their simplest form by performing operations such as combining like terms. It's like organizing a room by grouping similar items together to make it tidy and easy to navigate. In the context of algebra, we combine like terms to simplify expressions.

Let's take the provided exercise as an example. The process involves looking for terms that share the same combination of variables and their exponents. Once identified, like terms can be combined by adding or subtracting their numerical coefficients, based on the operation indicated between them. Always remember that the variables part must remain the same in the resulting term. The exercise provided shows this process in action, with the terms involving ab being combined to achieve a simpler expression.

Polynomial Arithmetic

Polynomial arithmetic is a branch of algebra that involves adding, subtracting, multiplying, and sometimes, dividing polynomials. Polynomials are algebraic expressions made up of multiple terms, much like the one we see in the exercise. The key to successfully performing polynomial arithmetic is understanding how to manipulate these terms correctly and ensuring like terms are combined appropriately.

In the step-by-step solution from the textbook, we perform polynomial arithmetic by first identifying and then combining like terms. The exercise shows this through the summation process of the coefficients of ab terms. Polynomial arithmetic often involves several steps, as we must address each operation present in the expression. It is the foundation for much of algebra and is essential for solving more complex mathematical problems.