Problem 106 - Intermediate Algebra

Question

Perform the indicated operations. $$ \left(6 a^{2} x^{3}-2 a x^{2}+3 a^{3}\right)+\left(-4 a^{2} x^{3}-2 a^{3}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Distribute the Signs

First, distribute the signs inside the parentheses. The expression given is: \( (6a^2x^3 - 2ax^2 + 3a^3) + (-4a^2x^3 - 2a^3) \). Since we need to add the two expressions, we simply remove the parentheses and distribute the '+' across them: \( 6a^2x^3 - 2ax^2 + 3a^3 - 4a^2x^3 - 2a^3 \).

Step 2: Combine Like Terms

Now, identify and combine like terms, which are terms with the same variable parts. The expression has the terms: \( 6a^2x^3 \) and \( -4a^2x^3 \) as like terms, \(-2ax^2\) as a standalone term, and \(3a^3\) and \(-2a^3\) as like terms. Combine them:- For \(a^2x^3\): \( (6 - 4)a^2x^3 = 2a^2x^3 \)- \(-2ax^2\) remains the same as there are no other like terms.- For \(a^3\): \( (3 - 2)a^3 = 1a^3 \) (or simply \( a^3 \)).

Step 3: Write the Final Simplified Expression

After combining all the like terms, the simplified expression is \( 2a^2x^3 - 2ax^2 + a^3 \). This is the result of the indicated operations on the given expressions.

Key Concepts

Like Terms

When dealing with polynomials, it is crucial to understand the concept of like terms. Like terms are terms that have exactly the same variables raised to the same powers. They can be combined through addition or subtraction, which simplifies the expression. In the given exercise, terms like \(6a^2x^3\) and \(-4a^2x^3\) are considered like terms, as they both contain \(a^2x^3\) with identical variable parts.
Recognizing like terms allows us to simplify complex polynomial expressions efficiently.

• Terms with identical variable parts and powers are like terms.
• Coefficients (numbers in front of variables) can be different.
• Only like terms can be combined.

Knowing how to spot like terms can make solving polynomial problems a smoother process.

Polynomial Operations

Performing operations on polynomials involves applying basic arithmetical operations such as addition, subtraction, multiplication, and sometimes division. In this context, adding polynomials means merging two or more polynomial expressions into a single, simplified one. The original exercise requires us to perform addition on two polynomials.
Whenever you see expressions with parentheses, it's a cue to begin by distributing the operations through each term.

• When adding polynomials, remove brackets if dealing with addition.
• Ensure to maintain the correct signs while combining terms.
• Tackle one term at a time to avoid confusion.

Polynomial operations require careful handling of terms to guarantee a correct outcome.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra where complex expressions are turned into simpler or more manageable ones. This typically involves combining like terms and eliminating unnecessary parentheses. In our example, the simplification involves a few strategic steps. After identifying like terms, we proceed to reduce the expression.

• Ensure all like terms are combined correctly.
• Cross-check to see if there are hidden possibilities for further simplification.
• Maintain accuracy in calculation at each step.

By mastering simplification, the final expression becomes not only easier to understand but also to work with in further calculations.