Problem 78 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ 8 j^{3}-9 j^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(-j^3\).

Step 1: Identify Like Terms

In this expression, identify the terms you will combine. The terms are \(8j^3\) and \(-9j^3\). Both are like terms since they have the same variable and exponent.

Step 2: Combine the Coefficients

Combine the coefficients of the like terms. This means calculating \(8 - 9\).

Step 3: Simplify the Expression

The result from combining the coefficients will give you \(8 - 9 = -1\). Therefore, the simplified expression is \(-1j^3\).

Key Concepts

Like Terms

In algebra, the concept of "like terms" is crucial for simplifying expressions. Like terms are terms that have the same variables raised to the same power. For instance, in the expression given: \(8j^3\) and \(-9j^3\), both terms share the same variable \(j\) and are raised to the third power \(j^3\). This categorizes them as "like" terms.
Understanding this helps in simplifying processes because like terms can be combined through operations such as addition or subtraction. This is crucial when you're trying to simplify a polynomial expression, ensuring that the expressions shrink to their simplest form. By only combining like terms, we maintain accuracy in our simplification process.

Combining Coefficients

When we combine like terms, we focus on their coefficients. Coefficients are the numerical parts of the terms. For instance, in \(8j^3\), \(8\) is the coefficient. Similarly, \(-9\) is the coefficient in \(-9j^3\).
To combine these, we perform basic arithmetic on their coefficients. In this case, you subtract \(9\) from \(8\), resulting in \(-1\).

• Step 1: Identify the coefficients of the "like terms".
• Step 2: Perform the arithmetic operation (addition or subtraction).
• Step 3: Apply the result to the common variable and power.

By focusing solely on the coefficients, we ensure that the variables are not altered during the process, leading to an accurate simplified form of the polynomial.

Polynomial Expression

A polynomial expression is made up of variables, coefficients, and exponents. Each separate part of a polynomial is called a "term." It's important to note that a polynomial can have multiple terms. For example, in the expression given, \(8j^3 - 9j^3\), there are two terms.
Polynomials can get complicated, especially when they have several terms and variables. Hence, simplifying them by combining the like terms helps to manage their complexity. This also makes further algebraic operations easier to perform.
By simplifying polynomial expressions, you not only make the math easier but also prepare the expression for more advanced operations, such as derivative calculations or factoring. The key is recognizing like terms to reduce the expression into its simplest form, ensuring it's as manageable and streamlined as possible.