Problem 23 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\left(4 m^{3}-3 m^{2}+5\right)+\left(-3 m^{3}-m^{2}+5\right)$$

Step-by-Step Solution

Answer

The sum is \(m^3 - 4m^2 + 10\).

Step 1: Rewrite the Expression

First, rewrite the expression by removing the parentheses, so you have all terms lined up: \(4m^3 - 3m^2 + 5 - 3m^3 - m^2 + 5\).

Step 2: Group Like Terms

Identify and group the like terms together. The like terms here are \(4m^3\) and \(-3m^3\), \(-3m^2\) and \(-m^2\), and the constant terms \(5\) and \(5\).

Step 3: Combine Like Terms

Combine the grouped like terms. For the \(m^3\) terms, \(4m^3 - 3m^3 = m^3\). For the \(m^2\) terms, \(-3m^2 - m^2 = -4m^2\). For the constant terms, \(5 + 5 = 10\).

Step 4: Write the Final Expression

Put the combined terms together in a single expression: \(m^3 - 4m^2 + 10\).

Key Concepts

Combining Like Terms

Combining like terms is an essential skill in simplifying and solving algebraic expressions. Like terms are those that have the same variables raised to the same power. This means that only the coefficients (the numbers in front of the variables) will differ. By combining them, you simplify the expression into a more manageable form.

To combine like terms effectively, follow these steps:

• Identify terms that have identical variable parts. For example, in the expression \(4m^3 - 3m^2 + 5 - 3m^3 - m^2 + 5\), \(4m^3\) and \(-3m^3\) are like terms.
• Find the sum or difference of the coefficients of these like terms. For instance, \(4m^3 - 3m^3\) simplifies to \(m^3\).
• Repeat this process for each group of like terms, including constant terms which are also considered like terms when they have no variable attached, like \(+5\) and \(+5\).

Remember, combining like terms helps to reduce a polynomial to its simplest form, making further calculations or transformations easier.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operation symbols. In algebra, understanding expressions is crucial as it forms the backbone of solving equations, modeling real-world scenarios, and performing operations.

Key features to note about algebraic expressions:

• They do not have an equal sign, unlike equations, thus representing parts of a whole that can undergo further manipulation.
• Expressions can range from simple (like \(5x\)) to complex (such as \(4m^3 - 3m^2 + 5\)).
• Each part of an expression is referred to as a term, and these terms are separated by plus or minus signs.

When working with algebraic expressions, particularly during operations like addition or subtraction of polynomials, ensure you properly line up the terms according to their degree and simplify by combining like terms. This streamlines computations and paves the way for solving the expression as part of a larger equation if needed.

Polynomials

Polynomials are a special kind of algebraic expression that consists of terms added or subtracted from each other, where each term is a product of a number (coefficient) and a variable raised to a non-negative integer power. Each polynomial is classified based on its degree, which is the highest power of the variable present.

Polynomials often form the basis of algebra's study because of their wide application and straightforward composition. Here are some important points:

• The degree can range from zero for constant polynomials (like \(7\)), to positive integers for linear (\(x + 5\)), quadratic (\(x^2 + 2x + 1\)), or higher.
• Polynomials can be added, subtracted, multiplied, and even divided under certain conditions.
• Working with polynomials often involves operations such as finding a sum or difference, just as seen in the original exercise: \(4m^3 - 3m^2 + 5\).

Simplifying a polynomial involves applying operations like combining like terms and arranging them in descending order based on their degree. This not only makes the work neat and organized, but it also provides the clarity needed for more complex operations or eventual equation solving.