Problem 27 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using the FOIL method and simplify. $$(x+3 y)(2 x-y)$$

Step-by-Step Solution

Answer

The simplified expression is \(2x^2 + 5xy - 3y^2\).

Step 1: Apply the FOIL Method

The FOIL method stands for First, Outer, Inner, Last. To apply FOIL, we'll multiply the terms of each binomial in the expression \((x + 3y)(2x - y)\) and sum the results: - First: Multiply the first terms from each binomial: \(x \cdot 2x = 2x^2\).- Outer: Multiply the outer terms: \(x \cdot (-y) = -xy\).- Inner: Multiply the inner terms: \(3y \cdot 2x = 6xy\).- Last: Multiply the last terms of each binomial: \(3y \cdot (-y) = -3y^2\).

Step 2: Combine the Like Terms

Now that we've applied FOIL, the expanded expression is: \[2x^2 - xy + 6xy - 3y^2\]Combine the like terms, which are the middle terms:- The terms \(-xy\) and \(6xy\) combine to give \(5xy\).

Step 3: Write the Simplified Expression

After combining the middle terms, our simplified expression is:\[2x^2 + 5xy - 3y^2\]

Key Concepts

Algebraic Expressions

When exploring algebra, you are likely to encounter algebraic expressions, which are combinations of numbers, variables, and operational symbols such as plus or minus. They represent a quantity in a general form rather than a specific number. - **Parts of an Algebraic Expression:** These typically consist of terms. Each term in an expression is a product of numbers and variables raised to a power. For example, in the expression \(3x^2 + 5xy - 2y\), there are three terms. - **Coefficient:** The numerical part of a term, such as 3 in \(3x^2\). - **Variable:** A symbol like \(x\) or \(y\) that represents numbers. - **Exponent:** A power that the variable is raised to, like 2 in \(x^2\).- **Adding and Subtracting Expressions:** When combining algebraic expressions, fundamentals such as combining like terms—terms with the same variables raised to the same power—play a crucial role. For instance, \(2x^2\) and \(5x^2\) could be added to make \(7x^2\).Understanding algebraic expressions is vital for exploring more complex concepts in algebra, such as solving equations and utilizing methods like FOIL.

Binomials

A binomial is a specific type of algebraic expression that contains exactly two terms. Each term is separated by a plus or minus sign. Binomials form the building blocks for many algebraic operations and factorization techniques. - **Examples of Binomials:** Expressions like \(x + 3y\) and \(2x - y\) are classified as binomials.- **Operations with Binomials:** When performing operations such as multiplication, the FOIL method is commonly used to expand expressions involving binomials.Binomials are not just foundational in algebra but also in analyzing polynomial functions and equations. They often precede more complex polynomial expressions, making mastery of operations like multiplying binomials essential to algebra.

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to a more concise form while retaining their original value and meaning. This often involves combining like terms and applying algebraic rules. - **Like Terms:** Terms that have identical variable parts. For example, \(-xy\) and \(6xy\) are like terms because they both contain \(xy\). These are combined to form \(5xy\).- **Using the FOIL Method:** An example of simplification can be seen when multiplying binomials. The FOIL method enables you to expand and then combine the terms to simplify, turning expressions like \((x + 3y)(2x - y)\) into \(2x^2 + 5xy - 3y^2\).- **Benefits:** Simplifying an expression makes it easier to understand and further manipulate or solve. It also prepares it for more complex operations, like solving or graphing.By practicing expression simplification, you gain a deeper understanding of how to handle complex algebraic problems efficiently and effectively.