Problem 17 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ \left(x^{3}+6 x^{2}-4 x+7\right)-\left(3 x^{2}+2 x-4\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( x^3 + 3x^2 - 6x + 11 \).

Step 1: Distribute the Negative Sign

First, we need to distribute the negative sign across the terms inside the second set of parentheses. This changes the expression from \( \left(3x^2 + 2x - 4\right) \) to \( -3x^2 - 2x + 4 \).

Step 2: Remove Parentheses

Rewrite the expression by removing the parentheses. After distributing the negative sign, the expression becomes: \( x^3 + 6x^2 - 4x + 7 - 3x^2 - 2x + 4 \).

Step 3: Combine Like Terms

Group and combine the like terms in the expression: - For the \(x^2\) terms: \(6x^2 - 3x^2 = 3x^2\) - For the \(x\) terms: \(-4x - 2x = -6x\) - For the constant terms: \(7 + 4 = 11\)This gives us the simplified expression: \( x^3 + 3x^2 - 6x + 11 \).

Step 4: Write Simplified Expression

The final simplified expression after performing all operations is: \( x^3 + 3x^2 - 6x + 11 \).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is like tidying up a room where everything is scattered. The goal is to make the expression as neat as possible by applying mathematical operations that don't change its overall value. In our original exercise, simplifying means performing actions that make it easier to work with:

• Distributing any multiplication symbols.
• Removing parentheses by distributing values contained within them.
• Combining parts that are alike (known as like terms).

The process of simplification involves rewriting the given expression in its simplest form. Once an expression is simplified, it is easier to understand and use in further calculations or solve in an equation.

Combining Like Terms

Combining like terms is essential in algebra to simplify expressions. Consider like terms as pieces from the same set. In the expression after the parentheses have been removed, terms are grouped based on their variables and degrees:

• The terms with the same degree, such as those containing the variable raised to the same power, are like terms.
• In our problem, the like terms are: \( x^2 \) terms, \( x \) terms, and constant numbers.
• For instance, combining \( 6x^2 \) and \( -3x^2 \) results in \( 3x^2 \) because you are adding and subtracting the coefficients directly.

Once like terms are combined, the expression becomes much simpler and is easier to handle in algebraic equations.

Distribution of Negative Sign

The distribution of a negative sign is a critical step in algebra, especially when simplifying expressions that involve subtraction. This action involves spreading the negative sign through the terms inside the parentheses.

• This means multiplying each term in the parentheses by \(-1\).
• In the original exercise, the expression \( 3x^2 + 2x - 4 \) becomes \( -3x^2 - 2x + 4 \) when the negative sign is distributed.
• Remembering to flip the signs is key here; positive terms become negative, and negative terms turn positive.

Performing distribution accurately ensures the rest of the expression simplification process can proceed smoothly.