Problem 105 - Intermediate Algebra

Question

Simplify each expression. $$3 x^{2}-\left(-2 x^{2}\right)-5 x^{2}$$

Step-by-Step Solution

Answer

The simplified expression is 0.

Step 1: Distribute the Negative Sign

The expression given is \(3x^2 - (-2x^2) - 5x^2\). The double negative sign in the middle turns into a positive. Rewrite the expression as \(3x^2 + 2x^2 - 5x^2\).

Step 2: Combine Like Terms

Combine the coefficients of the like terms. We have \(3x^2 + 2x^2 - 5x^2\). First, add \(3x^2\) and \(2x^2\) to get \(5x^2\). Now subtract \(5x^2\) from \(5x^2\).

Step 3: Perform the Arithmetic

Compute \(5x^2 - 5x^2\). This simplifies to \(0\). Therefore, the entire expression simplifies to 0.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to remove parentheses from an expression, making it easier to work with. In this specific exercise, we don't have any multiplication by individual terms across the parentheses, but we are applying the idea of distribution to manage signs effectively. Here, the expression

• \( 3x^2 - (-2x^2) - 5x^2 \).

involves double negatives. By applying the distributive property to the negative sign in front of the parenthesis, \( -(-2x^2) \) becomes

• \( +2x^2 \).

It is essential to correctly interpret and simplify these signs to avoid errors in calculations. Proper handling of signs and expressions is crucial in algebra as it lays the groundwork for more complex operations.

Combining Like Terms

Once we have simplified the signs in our expression, the next step is to combine like terms. "Like terms" are terms that have the same variable raised to the same power. In our case, all the terms in the expression are like terms because they all involve

• \( x^2 \),

making it easy to combine them. The expression

• \( 3x^2 + 2x^2 - 5x^2 \)

can be combined by simply adding and subtracting the coefficients,

• resulting in \( 5x^2 - 5x^2 \).

By effectively combining like terms, we simplify the expression further and reduce the potential for arithmetic mistakes. Keeping track of like terms is a crucial skill in algebra, simplifying complex expressions into manageable parts.

Polynomial Simplification

Polynomial simplification involves reducing an expression to its simplest form. This is achieved by applying operations like distributing and combining like terms. In our particular case, we've already reduced the expression to

• \( 5x^2 - 5x^2 \).

The arithmetic operation now becomes easily manageable, as

• \( 5x^2 - 5x^2 \)

leads directly to

• 0.

This result indicates complete cancellation between the terms. Simplifying polynomials is about making expressions neat and solvable. It involves using rules consistently to boil down expressions to their essence, allowing easier interpretation and problem-solving.