Problem 21 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (x-5)(x+3) $$

Step-by-Step Solution

Answer

The product of the two binomials \( (x-5)(x+3) \) is \( x^2 - 2x - 15 \).

Step 1: Distribute first term in the first binomial

First, distribute the first term in the first binomial i.e., \(x\), across \(x + 3\) in the second binomial. This gives: \(x(x) + x(3)\). This simplifies to \(x^2 + 3x\).

Step 2: Distribute second term in the first binomial

Distribute the second term in the first binomial, which is \(-5\), across \(x + 3\) in the second binomial. This gives: \(-5(x) -5(3)\). This simplifies to \(-5x -15\).

Step 3: Combine like terms

Now, add the results from Step 1 and Step 2, which gives: \(x^2 + 3x - 5x - 15\). Upon combining the like terms, specifically \(3x\) and \(-5x\), the result simplifies to \(x^2 - 2x - 15\).

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of constants, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In the exercise provided, \(x-5)(x+3)\) is an example of a binomial product, which means it contains two binomials being multiplied together. Each binomial consists of two terms, and in this case, we see terms including variables \(x\) and constants \(3\) and \(5\). When working with algebraic expressions, it's important to understand that operations should be performed following the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
To fully understand how to manage algebraic expressions, it's helpful to imagine these expressions as a recipe, where each ingredient must be combined in a specific order to create the desired result. Just like in a recipe, missing out on a single step or misplacing an arithmetic operation can lead to incorrect results. Therefore, mastering the technique of expanding and simplifying such expressions is crucial.

Distributive Property

The distributive property is a fundamental law of algebra that allows us to expand expressions where a term is multiplied by a binomial or polynomial. The property is represented as \(a(b + c) = ab + ac\), indicating that you multiply the term outside the parentheses by each term within the parentheses separately. In the given exercise, this property is utilized in both Step 1 and Step 2.
Let's visualize this concept with a real-world example: imagine you have a bag of apples to distribute equally to two friends. Rather than splitting the apples and then giving them separately to each friend, you would multiply the total number of apples by two. Similarly, with the distributive property, you 'distribute' the value of one term to all the terms inside the parentheses.

The Steps in Application

• Step 1: Distribute the term \(x\) across the binomial \(x + 3\), resulting in \(x^2 + 3x\).
• Step 2: Apply the property again with \(\-5\) to get \(\-5x - 15\).

By systematically applying the distributive property, students can expand and simplify algebraic expressions with precision.

Combining Like Terms

Once the distributive property has been applied, the next step is to 'combine like terms'. Like terms in algebra are terms that have the identical variable part, meaning they have the same variables raised to the same power. The coefficients of these terms can be different. In the product \(x^2 + 3x - 5x - 15\), the like terms are \(3x\) and \(\-5x\).
Think of combining like terms as grouping similar items. For instance, if you were organizing a fruit basket, you would place all the apples together and all the oranges together, not mix them up. Similarly, in algebra, we combine terms with \(x\) with other terms with \(x\) and constants with constants.

Combining Steps

• Add or subtract the coefficients of the like terms: \(3x - 5x = \-2x\).
• The constant terms (-15 in this case) remain unchanged as there are no similar constant terms to combine with.

After combining like terms, the expression simplifies to \(x^2 - 2x - 15\), completing the process. Learning to combine like terms is like sharpening a pencil—necessary for making the algebra 'writing' clearer and more precise.