Problem 20 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression, or explain why the expression cannot be simplified. $$14 x^{4}+x^{4}$$

Step-by-Step Solution

Answer

The simplified form of \(14x^{4} + x^{4}\) is \(15x^{4}\).

Step 1: Identify and Combine Like Terms

We first observe the expression \(14x^{4} + x^{4}\). Both terms include the variable \(x\) to the 4th power. Since these are 'like terms', they can be combined. This is done by adding or subtracting the numerical coefficients associated with these common variables. It can be misleading seeing 'x^4' with no obvious coefficient. But remember, when a term has no numerical coefficient written explicitly, it is assumed to be '1'. So, the expression actually reads '14x^4 + 1x^4'.

Step 2: Simplify the Expression

We then compute the sum of the coefficients 14 and 1, giving us \(15x^{4}\). Thus, \(14x^{4} + x^{4}\) simplifies to \(15x^{4}\).

Key Concepts

Understanding Like Terms

In algebra, 'like terms' are terms that have identical variable parts (including their exponents). For an expression to include like terms, the variables and their powers must be exactly alike. This is important because it allows us to combine these terms through addition or subtraction.

• Example: In the expression \(14x^4 + x^4\), both terms have the variable \(x\) raised to the 4th power, making them like terms.
• Example: In contrast, \(x^3\) and \(x^4\) are not like terms due to different exponents.

Like terms can be easily 'glued' together by combining their coefficients, simplifying the expression into a more manageable form. Recognizing like terms is a fundamental step in algebraic manipulation and significantly aids in the simplification process.

Process of Simplification

Simplification in algebra involves reducing an expression to its simplest form with no loss of information. This often includes combining like terms, cancelling out terms, and removing unnecessary parentheses.

• Step 1: Identify like terms, as managed in the previous section.
• Step 2: Combine like terms by adding or subtracting their coefficients.

For instance, in the expression \(14x^4 + x^4\), recognizing the like terms allows us to add their coefficients: 14 and 1. Hence, the expression simplifies to \(15x^4\).

Simplification makes equations easier to solve and understand. By reducing expressions, we make them cleaner and often more elegant.

Role of Coefficients

Coefficients are the numerical parts that multiply the variable in algebraic expressions. They play a critical role in understanding and manipulating expressions. For example, in \(14x^4\), the number 14 is the coefficient. It indicates how many times the base variable term \(x^4\) is counted.

• If a variable term has no coefficient explicitly written, like in \(x^4\), it is implied that the coefficient is 1.
• Understanding coefficients helps in adding, subtracting, and combining like terms effectively.

When simplifying expressions, we focus on the coefficients of like terms to find their sum or difference, enabling an efficient simplification process. Understanding and identifying coefficients quickly aids in reducing complex expressions into simpler forms.