Problem 106 - Intermediate Algebra

Question

Simplify each expression. $$8 x^{3}-x^{3}-\left(-2 x^{3}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(9x^3\).

Step 1: Identify Like Terms

First, identify the terms in the expression that have the same variable raised to the same power. In this expression, the like terms are: \(8x^3\), \(-x^3\), and \(-(-2x^3)\).

Step 2: Simplify Double Negative

The term \(-(-2x^3)\) simplifies by removing the double negative, which results in simply \(2x^3\). Thus, the expression becomes \(8x^3 - x^3 + 2x^3\).

Step 3: Combine Like Terms

Now, combine all the like terms by adding or subtracting their coefficients. Sum the coefficients: \(8 - 1 + 2 = 9\). Therefore, the simplified expression is \(9x^3\).

Key Concepts

Like Terms

In algebra, identifying like terms is a key step in simplifying expressions. Like terms have the same variable raised to the same power. For example, in our original expression, the terms \(8x^3\), \(-x^3\), and \(2x^3\) are like terms. Despite their different coefficients, they share the same variable \(x\) and the same exponent 3. Recognizing like terms allows us to focus only on the coefficients when combining the terms.

• Like terms have identical variable parts.
• Only the coefficients of like terms differ.
• Combining like terms makes equations easier to solve.

By gathering all like terms, you can simplify expressions significantly, making calculations more straightforward.

Double Negative

When you encounter a double negative in algebraic expressions, it’s important to simplify it correctly to avoid mistakes. A double negative means that two negative signs cancel each other out, effectively turning a negative into a positive.

• A negative of a negative becomes a positive.
• Apply this rule carefully to avoid errors in calculation.

In our case, the term \(-(-2x^3)\) simplifies to \(2x^3\). This step is crucial as it affects the sum of the coefficients in the expression. Remember, removing the double negative helps in presenting a cleaner equation, which is always preferred.

Combining Coefficients

Once like terms are identified and double negatives are simplified, the next step is combining the coefficients. This is done by simply adding or subtracting the numbers before the variables. In our exercise, after simplifying the terms, we had the expression \(8x^3 - x^3 + 2x^3\). Here's how the coefficients are combined:

• Add the positive and negative coefficients: \(8 - 1 + 2\).
• Calculate the sum: \(8 - 1 + 2 = 9\).
• The combination gives us a final coefficient of 9 for \(x^3\).

Combining coefficients is the final step in simplifying an expression involving like terms. It’s the crux of reducing an equation to its simplest form, highlighting the power of algebraic manipulation.