Problem 35 - Algebra 1: Concepts and Skills

Question

Use a vertical format to add or subtract. $$ \left(2 m-8 m^{2}-3\right)+\left(m^{2}+5 m\right) $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(-7m^2 + 7m -3\).

Step 1: Identify Like Terms

Like terms are terms that contain the same variables raised to the same power. In this case, \(2m\) and \(5m\) are like terms, as well as \(-8m^2\) and \(m^2\).

Step 2: Combine Like Terms

Combine the coefficients of the like terms by addition or subtraction. Thus, \(2m + 5m = 7m\) and \(-8m^2 + m^2 = -7m^2\). The \(-3\), being a constant, remains unchanged.

Step 3: Present the Final Simplified Expression

Combine all the terms together into a single algebraic expression: \(-7m^2 + 7m -3\) is the simplified form.

Key Concepts

Addition and Subtraction of Polynomials

The process of adding and subtracting polynomials involves combining terms that have identical variable parts, known as 'like terms.' When confronted with an addition task like
\[ (2m - 8m^2 - 3) + (m^2 + 5m) \],
arrange the polynomials in a vertical format to visually align like terms. This method simplifies the identification and combination of terms. Let's illustrate with a hierarchy:

• Separate and align terms with the same variable and exponent, one below the other.
• Next, add or subtract the coefficients of these like terms.
• Lastly, list all terms that are not like any others as they appear.

With the aligned terms, it becomes clear that you can directly combine them, leading to a simplified expression representing the sum or difference of the original polynomials.

Simplifying Algebraic Expressions

To simplify an algebraic expression means to make it as compact and straightforward as possible. This simplification often involves combining like terms and eliminating complexity where possible. The goal is to reduce the expression to its most basic form without changing its value or meaning.

• Identify terms that can be combined (like terms).
• Apply arithmetic operations to combine the coefficients of these terms.
• Present the expression in its simplest form with as few terms as possible.

It’s important not to overlook any term, including constants, as each plays a role in defining the simplified expression.

Like Terms in Algebra

In algebra, 'like terms' are the elements within an expression that contain the same variables and are raised to the same power. They are the building blocks that can be combined through addition or subtraction.

Characteristics of Like Terms

• They must have the same variable(s).
• Their exponents on corresponding variables must be identical.

When you come across terms like \(2m\) and \(5m\), they can be combined because they share the same variable \(m\), both to the first power. Similarly, \(-8m^2\) and \(m^2\) are like terms because they contain \(m\) squared. Recognizing like terms is a fundamental step in simplifying expressions.

Vertical Format Addition

Vertical format addition allows for a structured approach to combine polynomials. By stacking polynomials one on top of the other, it becomes easier to match and add like terms. This approach mitigates the risk of errors that might occur when trying to combine terms mentally or in a linear format. Here’s how to apply it:

• Write each polynomial above or below the other, making sure to align like terms vertically.
• Add or subtract the coefficients of these aligned terms.
• Bring down any terms that do not have like terms to combine with.

By doing this, you ensure a clear path to reach a correctly simplified expression.