Problem 91 - Beginning and Intermediate Algebra

Question

Write each expression without parentheses. $$ -(4 t+3 m) $$

Step-by-Step Solution

Answer

The expression without parentheses is \[\begin{equation} -4t - 3m \end{equation}\].

Step 1: Distribute the Negative Sign

The given expression is \[\begin{equation} -(4 t + 3 m) \end{equation}\]. The negative sign in front of the parenthesis means that we need to distribute otif to both terms inside the parenthesis.

Step 2: Multiply Each Term

Multiply the first term inside the parenthesis by -1: \[\begin{equation} -1 \times 4t = -4t \end{equation}\]. Then, multiply the second term inside the parenthesis by -1: \[\begin{equation} -1 \times 3 m = -3 m \end{equation}\].

Step 3: Combine the Terms

Combine the two results from Step 2 to get the final expression: \[\begin{equation} -4t - 3m \end{equation}\].

Key Concepts

Polynomials

In algebra, a polynomial is an expression made up of variables, coefficients, and exponents. For instance, in the expression \(4t + 3m\), both \(4t\) and \(3m\) are terms. Each term is a product of a coefficient (like 4 or 3) and a variable (like \(t\) or \(m\)). One important thing to note about polynomials is that they can be added, subtracted, and multiplied. Understanding the structure of polynomials, like seeing how terms are combined, is essential for many algebraic operations.

When you encounter problems involving polynomials, you often have to manipulate them by distributing signs, combining terms, or multiplying them. This exercise helps illustrate how to handle a negative sign to distribute it across terms in a polynomial.

Multiplication in Algebra

Multiplication in algebra often means distributing values across terms in an expression. In this exercise, we are distributing a negative sign through a set of terms.

Here's a step-by-step breakdown:

• Identify the terms inside the parenthesis: \(4t\) and \(3m\).
• Multiply each term by -1: \(-1 \times 4t = -4t\) and \(-1 \times 3m = -3m\).

This ensures that the sign is properly distributed across each term. Multiplying by -1 changes the sign of each term. It's crucial to remember that multiplication applies to both the coefficient and the variable linked in the term.

Combining Like Terms

Combining like terms is an essential skill in algebra to simplify expressions and solve equations more easily. Even though we're not explicitly combining like terms in this specific exercise, understanding how to do so is very valuable.

For instance, if we had an expression like \(-4t - 3m + 4t + 5m\), we'd have to combine like terms:

• Combine the terms with \(t\): \(-4t + 4t = 0\)
• Combine the terms with \(m\): \(-3m + 5m = 2m\)

This would give us the simplified expression: \(0 + 2m = 2m\).

Recognizing and combining like terms help reduce complex expressions to simpler forms, making it easier to reach solutions. It's like cleaning up and organizing your work in algebra.