Problem 33 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ x(x+1) $$

Step-by-Step Solution

Answer

The expression without parentheses is \(x^2 + x\).

Step 1: Identify the Term Outside the Parenthesis

The term outside the parentheses is 'x'. This term needs to be distributed across the expression within the parentheses.

Step 2: Distribute 'x' Across the Expression Within Parenthesis

Multiplying 'x' with 'x' yields \(x^2\), and multiplying 'x' with 1 yields 'x'. So, the expression can be written as \(x^2 + x\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. In simpler terms, these expressions are made up of constants (such as numbers) and variables (such as letters like 'x' or 'y') combined in various ways.
Variables represent unknown values and can vary, while constants have fixed numerical values.

• An example of a basic algebraic expression could be something like '3x + 5'.
• Here, '3' is a constant coefficient, 'x' is a variable, and '5' is a constant term.

Understanding algebraic expressions is crucial because they form the foundation of more complex equations and mathematical models. They allow you to generalize mathematical problems and solve them in various contexts once the values of the variables are known.

Parentheses

Parentheses in mathematics are symbols used to group parts of an expression or equation. They play a crucial role in determining the order of operations in a calculation, ensuring that certain operations are performed first.
In algebraic expressions, parentheses can alter the way we simplify or evaluate an expression.

• For instance, in the expression \((x+1)\), the parentheses signify that the operations inside should be performed together.
• When you have an operation outside the parentheses, such as multiplication, it applies to each term inside the parentheses.

In the given exercise, the expression \(x(x+1)\) uses parentheses to indicate that 'x' should be distributed to each term within the group \((x+1)\). This demonstrates their role in clarifying which calculations to perform first and consistently.

Simplifying Expressions

Simplifying expressions involves rewriting them in an equivalent, simpler form. This process often requires the application of algebraic laws or properties, such as the distributive property. The main goal is to make the expression easier to understand or solve.
The distributive property, specifically, is a crucial tool in simplifying expressions, especially those involving parentheses.

• The property states that for any numbers or variables \(a, b,\) and \(c\), the equation \(a(b+c) = ab + ac\) holds true.
• Applying this to \(x(x+1)\), you distribute 'x' across each term inside the parentheses to get \(x^2 + x\).

By breaking down each term using the property, the expression becomes easier to work with. Simplifying expressions is critical in algebra as it prepares equations for solving or further manipulation.