Problem 52 - Intermediate Algebra

Question

Perform the indicated operations. \(2 x^{3}-4 x^{3}\)

Step-by-Step Solution

Answer

The simplified expression is \(-2x^3\).

Step 1: Identify the Terms

Notice that the terms given are both cubic terms with the variable \(x\). The expression is \(2x^3 - 4x^3\).

Step 2: Combine Like Terms

Since both terms are alike (they have the same variable part \(x^3\)), you can combine them by performing the indicated arithmetic operation. Here, you subtract the coefficients: \(2 - 4\).

Step 3: Simplify the Coefficients

Calculate \(2 - 4\), which equals \(-2\).

Step 4: Write the Final Expression

Since the coefficients have been simplified to \(-2\), the combined expression with the variable \(x^3\) is \(-2x^3\).

Key Concepts

Understanding Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients. These are called "terms," and each term is a part of the expression. For example, in the expression \(2x^3 - 4x^3\), the terms are \(2x^3\) and \(-4x^3\). These terms are made up of:

• Coefficients: the numerical part, here they are 2 and -4.
• Variables: the letters, here it's \(x\).
• Exponents: the powers to which variables are raised, here both are raised to the power of 3.

Each part of a polynomial is key to understanding how to perform operations like addition, subtraction, and combining like terms. Polynomials can have different types of terms based on the exponent of the variables. This gives us categories like linear (degree 1), quadratic (degree 2), and cubic (degree 3). The understanding of polynomials is foundational in algebra as it helps in solving equations and understanding functions.

Combining Like Terms in Algebra

In algebra, "like terms" refer to terms that have the same variable raised to the same power. This is crucial for simplifying algebraic expressions. For instance, in \(2x^3 - 4x^3\), both terms are like terms because they contain \(x^3\). Before combining like terms, always ensure that:

• The variable part (including the exponent) is the same for each term.

Here's how you combine like terms:First, identify the coefficients of the terms. In this case, the coefficients are 2 and -4. Next, perform the arithmetic operation on the coefficients. For this example, you subtract 4 from 2, which equals -2. The variable and its exponent remain unchanged during this process. So, \(2x^3 - 4x^3\) simplifies to \(-2x^3\). This process shortens and simplifies expressions, making them easier to work with.

The Significance of Cubic Terms

Cubic terms refer to those in a polynomial where the variable is raised to the power of three, such as \(x^3\). These terms play a critical role when dealing with more complex polynomials, especially in equations involving third-degree polynomials. Here's what distinguishes cubic terms:

• They represent three-dimensional volume in geometry.
• They can have one or multiple roots.
• In physics, cubic equations often describe relationships involving volume or resistance.

When performing operations with cubic terms, it is important to remember that combining them follows the same rules as with lower-degree terms. The exponents do not change during addition or subtraction. Therefore, in examples like our expression \(2x^3 - 4x^3\), the result is \(-2x^3\). Recognizing the uniqueness of cubic terms allows for accurate manipulation and solution of algebraic problems involving higher powers.