Problem 125 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ a^{3}+2 a^{2}+4 a-2 a^{2}-4 a-8 $$

Step-by-Step Solution

Answer

The simplified expression is \(a^3 - 8\).

Step 1: Identify Like Terms

Look at the expression: \(a^3 + 2a^2 + 4a - 2a^2 - 4a - 8\). Identify like terms: the \(a^3\) term, \(a^2\) terms \((2a^2 - 2a^2)\), \(a\) terms \((4a - 4a)\), and the constant term \(-8\).

Step 2: Combine Like Terms

Now, combine the like terms together. Start with the \(a^2\) terms: \(2a^2 - 2a^2 = 0\) and \(a\) terms: \(4a - 4a = 0\). This leaves us with: \(a^3 - 8\).

Step 3: Verify Simplification

Check if the expression \(a^3 - 8\) can be simplified further. Since there are no like terms and no immediate factorizations possible, this is the simplest form.

Key Concepts

Understanding Like Terms

In algebra, the concept of "like terms" is crucial for simplifying expressions. Like terms are those that have exactly the same variables raised to the same power. Interestingly, the coefficients of these terms do not have to be identical.

Let's break it down with an example from the original expression: \( a^3 + 2a^2 + 4a - 2a^2 - 4a - 8 \).

In this expression:

• \( a^3 \) stands alone with no similar term.
• \( 2a^2 \) and \(-2a^2\) are like terms because they both involve \(a^2\).
• \( 4a \) and \(-4a\) are also like terms because they both involve \(a\).
• The term \(-8\) is a constant and has no like term in this expression.

Identifying like terms helps us to see which terms can be added together or combined. This is an initial step towards simplifying algebraic expressions efficiently.

Combining Like Terms

Once like terms are identified, the next step in simplification is combining them. This allows us to express the algebraic equation in a more compact form. Combining means adding or subtracting the coefficients of the like terms without changing the variable parts.

For example, take the like terms \(2a^2\) and \(-2a^2\):

• When combining these, we simply add the coefficients: \(2 + (-2) = 0\), resulting in \(0a^2\), which essentially means they cancel out.
• Similarly, for the terms \(4a\) and \(-4a\), we add the coefficients: \(4 + (-4) = 0\), resulting in \(0a\).

Consequently, these terms vanish from the expression, leaving us with any remaining terms. In this exercise, after combining, we simplify the expression to \(a^3 - 8\).

This process is vital in simplifying expressions, as it reduces the clutter and helps in focusing on the essential parts of the polynomial.

Simplifying Polynomial Expressions

Polynomials are algebraic expressions that consist of variables and coefficients, constructed using operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Simplifying polynomial expressions is about reducing the expressions to their most straightforward form.

The original expression we considered, \( a^3 + 2a^2 + 4a - 2a^2 - 4a - 8 \), is a polynomial expression. After identifying and combining like terms, this expression simplifies down to \( a^3 - 8 \).

When simplifying polynomials, always make sure to:

• Identify and combine all like terms.
• Ensure that the resulting expression is written in standard form, typically ordered by descending powers of variables.
• Check that no further simplification, such as factoring, is possible.

In our exercise, the result \( a^3 - 8 \) couldn't be simplified further as there were no more like terms and no obvious factorizations. This means the expression is already in its simplest form. Being able to simplify polynomials accurately is a fundamental skill in algebra, required for solving equations and other more advanced mathematical operations effectively.