Problem 47 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using the FOIL method, and simplify. \((2 r-5 s)(3 r-2 s)\)

Step-by-Step Solution

Answer

6r^2 - 19rs + 10s^2

Step 1: Applying FOIL Method

The FOIL method stands for First, Outer, Inner, Last. It's used to multiply two binomials. Let's break it down: 1. **First:** Multiply the first terms in each binomial: \(2r \cdot 3r = 6r^2\).2. **Outer:** Multiply the outer terms in the product: \(2r \cdot (-2s) = -4rs\).3. **Inner:** Multiply the inner terms: \(-5s \cdot 3r = -15rs\).4. **Last:** Multiply the last terms in each binomial: \(-5s \cdot (-2s) = 10s^2\).

Step 2: Combine Like Terms

Now that we've applied the FOIL method, combine the resulting terms:1. Combine \(6r^2\).2. Combine the like terms, \(-4rs\) and \(-15rs\), to get \(-19rs\).3. Combine \(10s^2\).The expression becomes: \[6r^2 - 19rs + 10s^2\].

Step 3: Simplify the Expression

After combining like terms, we verify that no further simplification is possible. The simplified expression is:\[6r^2 - 19rs + 10s^2\].

Key Concepts

Algebraic Expressions

In algebra, an expression comprises numbers, variables, and operations, like addition or subtraction, without an equal sign. Algebraic expressions can involve many terms, and each term can be a number, a variable, or a product of numbers and variables. For instance, expressions such as \( 3x + 2y \) or \( 5ab^2 \) show different combinations.
These expressions are used to represent real-world quantities or solve equations. Understanding how to manipulate and operate on algebraic expressions is key to solving algebra problems.
When working with these expressions, one important skill is being able to identify and perform operations on them efficiently, like in the given exercise.

Binomials

A binomial is a type of algebraic expression containing exactly two terms. These terms are typically joined by either a plus or minus sign. For example, both \( 2r - 5s \) and \( 3r + x \) are binomials.
Binomials often appear in many different types of algebraic problems, from simple calculations to complex equations. In the given exercise, we are asked to multiply the two binomials, \((2r-5s)\) and \((3r-2s)\).
We use specific techniques, like the FOIL method, to handle operations involving binomials, especially when multiplying.

Like Terms

Like terms are terms within an algebraic expression that have identical variable parts. This means they can be combined through addition or subtraction. For example, \(3x\) and \(5x\) are like terms and can be combined to become \(8x\).
The skill of recognizing and combining like terms is crucial in simplifying expressions and solving equations. In the step-by-step solution for our exercise, we identified like terms, such as \(-4rs\) and \(-15rs\), and combined them to get \(-19rs\).
This process helps to simplify the expression and makes it easier to handle.

Simplification of Expressions

Simplification in algebra involves rewriting an expression in its simplest form, often to make calculations easier. This involves combining like terms, eliminating brackets, and performing arithmetic operations.
In the context of the original problem, after applying the FOIL method and obtaining terms like \(6r^2\), \(-4rs\), \(-15rs\), and \(10s^2\), we simplified by combining like terms, arriving at \(6r^2 - 19rs + 10s^2\).
Successfully simplifying an expression can reveal insights about the expression's underlying structure and potential factors, making further mathematical operations more straightforward.