Problem 53 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\(5 x(x+1)-3 x(x+1)\)

Step-by-Step Solution

Answer

The simplified product is \(2x^2 + 2x\).

Step 1: Distribute Multiplication

In the first step, distribute the multiplication within the parentheses for both terms. It will give us \(5x^2 + 5x - 3x^2 - 3x\).

Step 2: Combine Like Terms

In the second step, combine the like terms. The terms \(5x^2\) and \(-3x^2\) combine to give \2x^2\ and the terms \(5x\) and \(-3x\) combine to give \(2x\). Thus the final result of simplification is \(2x^2 + 2x\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions by distributing multiplication over addition or subtraction within parentheses. Think of it as spreading the multiplication over each term inside the parentheses.
In our given problem, we have two expressions where distribution is necessary:

• First, for the term \(5x(x+1)\), we apply the distributive property by multiplying \(5x\) with \(x\) and then \(5x\) with \(1\). This results in \(5x^2 + 5x\).
• The same process applies to the term \(-3x(x+1)\). We multiply \(-3x\) with \(x\) and then \(-3x\) with \(1\), yielding \(-3x^2 - 3x\).

This expansion makes the expression easier to work with and sets the stage for combining like terms.

Combining Like Terms

Combining like terms is the process of simplifying an expression by merging terms that have the same variables and powers. This helps to consolidate the expression into a more manageable form.
In our solution, once the distribution is complete, the expression becomes \(5x^2 + 5x - 3x^2 - 3x\).
To combine like terms:

• Look for terms with the same variable and exponent. In this case, we have the \(x^2\) terms, namely \(5x^2\) and \(-3x^2\), which combine to form \(2x^2\). This is because they both share the same base and exponent.
• Next, combine the \(x\) terms: \(5x\) and \(-3x\). Adding these gives us \(2x\).

By adding these results together, we simplify the expression to \(2x^2 + 2x\). This is now far simpler to work with or understand.

Polynomial Simplification

Polynomial simplification is the process of reducing a polynomial to its simplest form, making it easier to understand and solve. This often involves using both the distributive property and combining like terms.
After applying the distributive property and combining like terms, we derive the simplified polynomial \(2x^2 + 2x\).
This process is important because:

• Simplified polynomials are easier to interpret and use in further calculations or real-world applications, like modeling a situation.
• A simplified expression avoids unnecessary complexity, making further algebraic manipulation or solving equations straightforward.

In summary, polynomial simplification is a valuable tool in algebra that enhances clarity and efficiency in mathematical operations.