Problem 23 - Introductory Algebra for College Students

Question

In Exercises \(15-28,\) simplify each algebraic expression, or explain why the expression cannot be simplified. $$27 x^{3}-26 x^{3}$$

Step-by-Step Solution

Answer

The simplified expression is \(x^{3}\).

Step 1: Identify Like Terms

In the given expression \(27 x^{3}-26 x^{3}\), both terms are like terms because they both contain the same variable \(x^{3}\).

Step 2: Combine Like Terms

Subtract the coefficient of the second term from the coefficient of the first term. This is done like this: \(27 - 26 = 1\).

Step 3: Write The Simplified Expression

The simplified expression will be the result from the previous step followed by the common variable which is \(x^{3}\). Therefore, the simplified expression is \(1x^{3}\). But since the coefficient is 1, it's not necessary to write it. The simplified expression then becomes \(x^{3}\).

Key Concepts

Like Terms

In algebra, the concept of "like terms" is essential for simplifying expressions. Like terms are terms that have the exact same variables raised to the same powers. For instance, consider the terms \(27x^3\) and \(-26x^3\). Both of these terms have the variable \(x\) raised to the third power and are considered like terms. The coefficients (numbers in front) can be different, but the variable part must be the same.
Understanding this concept makes it easier to identify parts of an equation that can be simplified through addition or subtraction.

• Like terms have identical variable components.
• They enable simplification in algebraic expressions.
• Recognizing like terms is crucial before combining or simplifying them.

Combining Like Terms

Combining like terms is a fundamental step in simplifying algebraic expressions. To combine them, you perform arithmetic operations (addition or subtraction) on their coefficients while keeping the variable part unchanged.
For example, in the expression \(27x^3 - 26x^3\), we identify the like terms and subtract their coefficients, resulting in \((27 - 26)x^3 = 1x^3\). When the coefficient is 1, it is usually omitted, so the expression simplifies to \(x^3\).
Here are the steps to combine like terms effectively:

• Identify terms with the same variables and exponents.
• Add or subtract the coefficients of these terms.
• Rewrite the expression with the simplified coefficients.

Adopting this method helps in reducing expressions to their simplest forms.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators (like \(+\), \(-\), \(\times\), and \(\div\)). These expressions do not have equal signs, which differentiates them from equations. For example, \(27x^3 - 26x^3\) is an algebraic expression.
Understanding how to work with algebraic expressions is essential because they form the basis of algebra and are used to denote real-world relationships and operations.

• Consist of numbers, variables, and operations.
• Can be simplified to more manageable forms.
• Used to represent mathematical ideas without an equality sign.

Mastering algebraic expressions opens up the ability to solve complex algebraic equations.

Coefficient

In algebra, a coefficient is the numerical factor that multiplies a variable in a term. It appears in front of the variable component. For instance, in the terms \(27x^3\) and \(-26x^3\), 27 and -26 are coefficients.
The coefficient indicates how many times the variable is used. Understanding coefficients is key when combining like terms because we perform operations on these numbers.
Here’s how coefficients work:

• They multiply with the variables in a term.
• Crucial for combining like terms through arithmetic operations.
• When the coefficient is 1, it is often not shown (e.g., \(1x^3\) is written as \(x^3\)).

Knowing how to handle coefficients allows for effective expression simplification and manipulation.