Question

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

Short answer

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

Step-by-step solution

Identify Like Terms

Like terms are those which have exactly the same variable raised to the same exponent. Here, \(8 y^{2}\) and \(-3 y^{2}\) are like terms, and \(2\) and \(5\) are like terms because they are constant terms without any variables.

Add/Subtract Like Terms

Add the coefficients of the like terms (treat subtraction as addition of a negative). So, for \(8 y^{2}\) and \(-3 y^{2}\), the resultant is \(5 y^{2}\). Similarly, for the terms \(2\) and \(5\), the resultant is \(7\). Remember, the variable part remains the same.

Combine the Results

Append both results from step 2 together to form the final result. Here, the result of summing the \(y^{2}\) terms comes first (by convention, higher power terms come before lower power terms), followed by the result of summing the constant terms.

Key concepts

Like Terms

Understanding 'like terms' is essential when working with polynomials. Like terms are components of an expression that have the exact same variables raised to the same power. For example, in the given exercise, 8y^2 and -3y^2 are like terms because they both contain the variable y squared. Similarly, constants without variables, such as 2 and 5, are also considered like terms. Recognizing like terms allows us to simplify expressions by combining them accurately, which is a foundational skill in algebra.

Additionally, it's worth noting that like terms are not restricted to simple variables and can include more complex expressions involving products or quotients, as long as the variable portion is identical. For instance, 2xy^3 and -7xy^3 are like terms, while 2xy^3 and 2xy^2 are not, because the exponents on y differ.

Combining Like Terms

Once like terms have been identified, the next step is to combine them through addition or subtraction. This process is known as combining like terms. To do this, we keep the variable part the same and only add or subtract the coefficients. For instance, in our exercise, we combine 8y^2 and -3y^2 by adding their coefficients (8 and -3) to get 5y^2. Similarly, we combine the constants 2 and 5 to get 7.

It's important to note that only the numerical coefficients change when combining like terms; the variables and their exponents remain unchanged. Also, subtraction can be viewed as adding a negative, which can sometimes simplify the process. Combining like terms effectively simplifies expressions and makes further algebraic operations more manageable.

Polynomial Operations

Polynomial operations include addition, subtraction, multiplication, and division of polynomial expressions. When adding or subtracting polynomials, we focus on combining like terms, as demonstrated in the exercise. It's essential to align each term properly according to its degree and only perform operations with like terms.

For multiplication, we use the distributive property, often referred to as the FOIL method when dealing with binomials. Division of polynomials can be more complex, sometimes requiring long division or synthetic division methods. Understanding the behavior and structure of polynomials is key to mastering these operations, which are commonly applied in various areas of mathematics, including calculus and algebraic geometry.

Vertical Format Addition

Vertical format addition is a helpful method for adding polynomials. It involves writing each polynomial one underneath the other, aligning the like terms vertically. This visual arrangement makes it convenient to identify and combine like terms. In our exercise, we would write 8y^2 + 2 above -3y^2 + 5, with each corresponding term positioned directly above or below its counterpart.

After aligning the like terms, proceed to add the coefficients vertically. This method not only helps in organizing your work and avoiding mistakes but also prepares you for more advanced operations involving polynomials, such as polynomial long division. Beginners find vertical addition particularly useful because it clearly demonstrates the process of combining like terms and simplifying expressions step by step.