Problem 45 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(2x^2 + 5xy - 3y^2\)

Step 1: First Terms

Multiply the first terms of each binomial: \(x\) from \((x + 3y)\) and \(2x\) from \((2x - y)\). This gives us:\[x \cdot 2x = 2x^2\]

Step 2: Outside Terms

Multiply the outside terms: \(x\) from \((x + 3y)\) and \(-y\) from \((2x - y)\). This gives us: \[x \cdot (-y) = -xy\]

Step 3: Inside Terms

Multiply the inside terms: \(3y\) from \((x + 3y)\) and \(2x\) from \((2x - y)\). This gives us: \[3y \cdot 2x = 6xy\]

Step 4: Last Terms

Multiply the last terms: \(3y\) from \((x + 3y)\) and \(-y\) from \((2x - y)\). This gives us: \[3y \cdot (-y) = -3y^2\]

Step 5: Combine Like Terms

Combine all the results from the previous steps: \(2x^2 + (-xy) + 6xy + (-3y^2)\). Simplify by combining like terms: \[-xy + 6xy = 5xy\]. So, we have: \[2x^2 + 5xy - 3y^2\]

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of algebra. They consist of variables, constants, and the operations that combine them, such as addition, subtraction, multiplication, and division.
A variable is a symbol, often a letter, that represents a number that can change. A constant is a fixed number. For example, in the expression \( x + 3y \), \( x \) and \( y \) are variables, while 3 is a constant.
When working with algebraic expressions, we can manipulate these symbols according to the rules of arithmetic to simplify or transform them. Understanding how to handle algebraic expressions is crucial when performing operations, such as the FOIL method.
By mastering the manipulation of these expressions, you can solve equations and find unknown values.

Binomials

A binomial is a type of algebraic expression that includes exactly two terms. These terms are usually connected by either a plus or minus sign.
For example, in the expression \((x + 3y)\), we have a binomial consisting of the two terms \(x\) and \(3y\). Binomials are often involved in multiplication problems, where you may be asked to find the product of two binomials.
Understanding binomials is key to applying the FOIL method, an effective strategy for multiplying two binomials.

• The acronym stands for: First, Outside, Inside, Last
• It represents the order of multiplication of the terms in the binomials

Grasping this concept will help you correctly expand expressions and is an essential skill for higher-level math problems.

Like Terms

Like terms are terms that have the same variable components raised to the same powers. They can be combined through addition or subtraction because they represent the same kinds of quantities.
For instance, in the expression \(-xy + 6xy\), both terms are like terms because they both contain the variable \(xy\). Combining like terms simplifies expressions by reducing the number of terms.
In the step-by-step solution we provided, after performing the multiplication, we found two like terms: \(-xy\) and \(6xy\). By combining these, we arrived at \(5xy\). Recognizing and combining like terms is crucial to simplifying expressions effectively.

Simplification

Simplification in algebra involves reducing an expression to its most concise form while keeping its value the same. This often involves combining like terms, as was done in the last step of our original solution.
In the expression \(2x^2 + 5xy - 3y^2\), each term is distinct and cannot be combined further.

• \(2x^2\) remains separate from the other terms because it involves a different power or set of variables
• \(5xy\) and \(-3y^2\) also remain distinct

Simplification makes expressions easier to understand and work with, especially for solving equations or further manipulations. Understanding how to simplify expressions is a foundational skill that enhances clarity and precision in mathematical problem-solving.