Problem 63 - Algebra 1: Concepts and Skills

Question

Simplify the expression by combining like terms if possible. If not possible, write already simplified. $$ -8 m-m^{2}+2 m $$

Step-by-Step Solution

Answer

The simplified expression is \(-6m - m^{2}\)

Step 1: Identify Like Terms

We recognize the like terms in the expression. These are \(-8m\) and \(2m\) as they both have the variable 'm' raised to the power of 1.

Step 2: Combine Like Terms

Next we add or subtract the coefficients of these two terms: \(-8 + 2 = -6\). We are left with \(-6m.\)

Step 3: Rewrite the Expression

We rewrite the initial expression by replacing the like terms with their sum: \(-6m -m^{2}\). This is the simplified form of the expression.

Key Concepts

Simplifying Expressions

Simplifying expressions involves making a math expression as simple as possible. It means rewriting the expression in a way that is easier to handle or understand. The goal is to eliminate any redundant or troublesome parts of the expression. This often involves combining like terms.

When we simplify, we aim to make calculations easier and more efficient. Each small step we take should reduce the complexity of the expression. It doesn't mean changing the value, but rather giving us an equivalent expression that's easier to read.

• Clean up the expression by removing any unnecessary components.
• Make complex expressions look simpler without altering their value.
• Enable easier computation by reducing cumbersome elements.

Always remember, simplifying does not change what the expression means or represents. It only makes it easier to work with.

Identifying Like Terms

Before we can simplify an expression, it's crucial to identify like terms. Like terms are terms that have the same variables raised to the same power. They can vary in their coefficients—the numbers in front of these variables—but their variable parts need to match.

For example, in an expression involving terms like \(-8m\) and \(2m\), we say these terms are 'like' because they both include the variable \(m\) raised to the same power, which is 1 in this case. However, a term like \(-m^2\) is not like these because the power of \(m\) is different.

• Focus on the variable and its power to determine likeness.
• Don't worry about the coefficient when identifying like terms.
• Keep terms together that share the same variable and power.

Identifying like terms makes it easier to combine them, which is a key step in simplifying expressions.

Polynomial Expressions

A polynomial expression consists of terms that are made up of variables and coefficients combined through addition, subtraction, and multiplication. Each term in these expressions can have one or more variables raised to a non-negative integer power.

Polynomials must be fully explored to manage simplification and operations properly:

• Understand how many terms exist within the polynomial.
• Take note of the powers and variables that make up these terms.
• Recognize that polynomials can have a variety of terms, from constants (just numbers) to more complex terms with variables.

In the context given, \(-8m - m^2 + 2m\) is a polynomial expression with three distinct terms. The key to working with such expressions lies in identifying like terms and simplifying them when possible. Aligning all terms appropriately is essential to gaining a simplified form.