Problem 62 - Intermediate Algebra

Question

Multiply. See Example 4 $$(2 t+5)(-2)$$

Step-by-Step Solution

Answer

The expression \((2t + 5)(-2)\) simplifies to \(-4t - 10\).

Step 1: Distribute the First Term

Start with the expression \((2t + 5)(-2)\). Distribute the term \(-2\) to each term inside the parentheses. First, multiply \(-2\) by \(2t\):\[-2 imes 2t = -4t\].

Step 2: Distribute the Second Term

Next, multiply \(-2\) by the constant term \(5\) inside the parentheses:\[-2 imes 5 = -10\].

Step 3: Combine the Results

Combine the results of the two products to form the final expression. The expression \((2t + 5)(-2)\) simplifies to:\[-4t - 10\].

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. In our exercise, the expression is \(2t + 5\), which includes:

• \(2t\), where \(2\) is a coefficient and \(t\) is a variable.
• A constant term, \(5\).

These expressions are fundamental in algebra and provide a way to represent real-world problems using symbols. They help in carrying out operations, such as addition, subtraction, and multiplication. Understanding algebraic expressions is the first step to manipulating them effectively.

Mastering Multiplication of Terms

Multiplying terms within an expression is all about applying the distributive property. This property allows you to multiply a single term by each term inside a set of parentheses. In our example, the term \(-2\) is multiplied by each term in \(2t + 5\). Here’s how it works:

• First, multiply \(-2\) by \(2t\), which gives \(-4t\).
• Then, multiply \(-2\) by \(5\), resulting in \(-10\).

This step-by-step multiplication ensures every term is accounted for, allowing accurate simplification of the expression. Mastering this multiplication process is essential for solving more complex algebraic problems.

Introduction to Polynomials

A polynomial is an algebraic expression that can have multiple terms, including constants, variables, and exponents. The given expression \(2t + 5\) is a binomial, a type of polynomial with exactly two terms. Polynomials can range from simple, like in our example, to more complex with several variables and higher powers. Knowing how to work with polynomials is crucial because:

• They can model a wide range of real-world problems.
• They form the basis for advanced mathematical concepts.

By understanding polynomials, their structure, and how they interact through operations like multiplication, you can tackle a variety of algebraic challenges more effectively.