Problem 49 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 3 z^{2}-z+3 z^{3} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \(3 z^{2} - z + 3 z^{3}\).

Step 1: Identify the terms with the same exponent

In the expression \(3 z^{2} - z + 3 z^{3}\), there are three terms, but none of them have the same exponent to combine. The terms are: 1. \(3 z^{2}\) 2. \(-z\) (which can also be written as \(-1 z^{1}\)) 3. \(3 z^{3}\)

Step 2: Combine the terms

Because the terms cannot be combined, the expression is already in its simplest form. Therefore, the simplified expression is: $$ 3 z^{2} - z + 3 z^{3} $$

Key Concepts

Algebraic Terms

Understanding algebraic terms is the foundation to mastering algebra. These terms are the building blocks of algebraic expressions and equations, representing quantities that can vary. An algebraic term consists of numbers, variables, or a combination of both, sometimes separated by arithmetic operations such as addition or subtraction.

For example, in the expression \( 3 z^{2} - z + 3 z^{3} \), there are three distinct algebraic terms: \( 3 z^{2} \), \( -z \), and \( 3 z^{3} \). Each term is made up of a coefficient and a variable raised to an exponent. The coefficient is the numerical factor, while the variable part represents an unknown quantity, which in this case is \( z \). The exponent denotes how many times the variable is multiplied by itself. Recognizing and distinguishing between these terms is the first step in simplifying algebraic expressions.

Algebraic Simplification

Algebraic simplification is a process that involves reducing expressions to their simplest form. The aim is to make the expression easier to understand or work with, by combining like terms, applying the distributive property, and eliminating any unnecessary complexity.

To simplify an expression, one must look for terms that have the same variables raised to the same power, as these can be combined by addition or subtraction—this is known as combining like terms. However, when an expression, such as \( 3 z^{2} - z + 3 z^{3} \), has no like terms, it is already as simple as it can be. In the provided example, none of the terms have the same exponent to be combined.

Combine Like Terms

In other instances, if you had terms like \( 2x \) and \( 3x \), you could combine them to \( 5x \) because they have the same variable to the same power. Simplifying expressions not only makes them more visually understandable but also prepares them for further algebraic manipulations.

Exponents in Algebra

Exponents play a crucial role in algebra, symbolizing repeated multiplication of the same number or variable. They are written as a small number, known as the exponent, to the upper right of the base number or variable. The base tells us what number is being multiplied, and the exponent indicates how many times it is multiplied by itself.

For example, \( z^{3} \) represents \( z \times z \times z \) and \( 4^{2} \) represents \( 4 \times 4 \). In algebraic expressions like \( 3 z^{2} - z + 3 z^{3} \), correct interpretation of exponents is vital to understanding and simplifying the expression.

Rules of Exponents

Fundamental rules such as product of powers, power of a power, and power of a product can be used to manipulate expressions involving exponents, allowing for further simplification where possible. However, in our example, the terms have different exponents and therefore must be left as they are. Recognizing when terms with exponents can and cannot be combined is a key aspect of algebraic simplification.