Problem 1 - Elementary Algebra

Question

For Problems \(1-10\), find the indicated products by applying the distributive property; for example, $$ \begin{aligned} (x+1)(y+5) &=x(y)+x(5)+1(y)+1(5) \\ &=x y+5 x+y+5 \end{aligned} $$ $$ (x+2)(y+3) $$

Step-by-Step Solution

Answer

The product is \(xy + 3x + 2y + 6\).

Step 1: Apply the Distributive Property

To find the product \((x+2)(y+3)\), we use the distributive property, which states that \(a(b+c) = ab + ac\). Here, we distribute each term in the first binomial \((x+2)\) across each term in the second binomial \((y+3)\).

Step 2: Multiply the First Term of the First Binomial

First, multiply the term \(x\) by each term in the second binomial \((y+3)\). This gives us \(x(y) + x(3) = xy + 3x\).

Step 3: Multiply the Second Term of the First Binomial

Next, multiply the term \(2\) by each term in the second binomial \((y+3)\). This yields \(2(y) + 2(3) = 2y + 6\).

Step 4: Combine the Products

Combine all the products we obtained from the distributive steps: \(xy + 3x + 2y + 6\). These represent the expanded form obtained by distributing each term.

Key Concepts

Binomial Expansion

When dealing with algebraic expressions, we often come across binomials. A binomial is simply an algebraic expression containing two distinct terms. For example, \(x+2\) and \(y+3\) are binomials. Binomial expansion is the process of multiplying these expressions together to form a polynomial. In the exercise provided, the goal is to find the product of \(x+2\) and \(y+3\).
When expanding binomials, you need a method that ensures each term in the first binomial is multiplied by each term in the second. This is where the distributive property becomes useful. The product of two binomials results in four terms, which can often be combined to simplify the expression. Understanding binomial expansion helps in simplifying complex expressions, solving equations, and aids in further mathematical problem solving.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition and multiplication. They form the basis of algebra, allowing us to represent and solve real-world and theoretical problems in mathematics.
An algebraic expression could be as simple as \(x + 2\) or more complex involving multiple terms and variables. The exercise \( (x+2)(y+3) \) is an example of using algebraic expressions in binomial multiplication.

• They allow us to generalize arithmetic, providing a framework for representing patterns mathematically.
• Algebraic expressions can be simplified or rearranged using various algebraic rules and properties, such as the distributive property.
• They are crucial for forming equations and functions, which model relationships and changes.

Understanding how to manipulate algebraic expressions is key to mastering algebra and is essential for higher-level math, including calculus and beyond.

Distributive Law Application

The distributive property, one of the core properties of arithmetic, is crucial when working with algebraic expressions like binomials. It's defined as \(a(b+c) = ab + ac\). This law allows us to break down expressions and make them more manageable.
In the problem \( (x+2)(y+3) \), the distributive property was applied step-by-step:

• First, the term \(x\) from \(x+2\) was distributed to multiply each term in \(y+3\), producing \(xy + 3x\).
• Second, the term \(2\) from \(x+2\) was also distributed to each term in \(y+3\), resulting in \(2y + 6\).
• Finally, these products were combined to express the expanded form: \(xy + 3x + 2y + 6\).

Using the distributive property simplifies the process of expanding binomials and is a foundational concept for understanding more complex algebraic operations. Familiarity with this property allows students to confidently tackle arithmetic with variables and prepare for advanced mathematical studies.