Problem 36 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The result of the addition is \(5y^{2} + 7\).

Step 1: Identify like terms

Like terms are terms that involve the exact same variable raised to the same power. Here, the like terms are \(8y^{2}\) and \(-3y^{2}\), and \(2\) and \(5\).

Step 2: Perform addition on like terms

We add together each pair of like terms. This gives \(8y^{2} - 3y^{2} = 5y^{2}\) for the y-squared terms, and \(2 + 5 = 7\) for the constant terms.

Step 3: Combine resulting terms

After adding like terms, we can combine the results to form a single polynomial expression. Here, we put together \(5y^{2}\) and \(7\) to obtain our final answer \(5y^{2} + 7\).

Key Concepts

Understanding Like Terms

In algebra, like terms are crucial for simplifying expressions easily and correctly. Like terms contain the same variables raised to the same power, allowing them to be added or subtracted. For example, in the expression \(8y^2 - 3y^2\), both terms are like terms because they involve \(y^2\).
Recognizing like terms enables you to:

• Streamline your calculations
• Avoid mistakes by only combining compatible terms
• Simplify expressions efficiently

Identifying these terms requires careful attention, especially in complex expressions.

Using Vertical Format for Addition

Using the vertical format for adding or subtracting polynomials can simplify the process significantly. This method involves lining up like terms vertically, similar to traditional arithmetic.
To apply vertical format:

• Align like terms in columns
• Add or subtract the terms column by column
• Simplify the result for a clean final expression

This technique not only ensures accuracy but also helps visualize the components being combined, making it a preferred choice for many students.

What Are Polynomials?

Polynomials are expressions consisting of variables and coefficients, involving operations like addition, subtraction, multiplication, and non-negative integer exponents.
Key characteristics of polynomials:

• Contain one or more terms
• Each term includes a variable raised to an exponent and has a coefficient
• Expressions like \(5y^2 + 7\) are examples of polynomials

Understanding the structure of polynomials is the first step towards mastering algebraic operations.

Exploring Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation signs like \(+\) and \(-\). They can be simple like \(x+2\), or complex, involving multiple terms and operations. These expressions form the building blocks of algebra.
Features of algebraic expressions include:

• Variables that can assume various numeric values
• Mathematical operations that define relationships within the expression
• The potential to solve real-world problems when manipulated correctly

Being comfortable with these expressions is essential for algebra, calculus, and beyond, as they form the foundation for more advanced topics.