Problem 65 - Algebra 1

Question

COMBINING LIKE TERMS Apply the distributive property. Then simplify by combining like terms. $$ 7 x-3 x(x+1) $$

Step-by-Step Solution

Answer

-3x^2 + 4x

Step 1: Apply the Distributive Property

Applying the distributive property to -3x(x+1) gives -3x^2 - 3x. Therefore the new expression becomes 7x - 3x^2 - 3x.

Step 2: Combine Like Terms

After applying the distributive law, now combine the like terms which are 7x and -3x together. So, the expression further simplifies to 4x - 3x^2.

Step 3: Rearrange the Terms

Generally, expressions are written in decreasing powers of x. Therefore, rearranging the terms, the final simplified expression is -3x^2 + 4x.

Key Concepts

Distributive Property

When dealing with algebraic expressions, one fundamental tool is the distributive property. This property enables us to simplify expressions by multiplying a single term by each term inside parentheses. In mathematical terms, the distributive property is expressed as:

• \( a(b+c) = ab + ac \)

This means you take the number or variable outside the parentheses and multiply it by each term within. For example, in the given expression \(-3x(x+1)\), we apply the distributive property by multiplying \(-3x\) with each term inside the parentheses. Hence, it becomes \(-3x^2 - 3x\).
By distributing the multiplier (\(-3x\)) across the terms within the parentheses, we can simplify complex expressions into more manageable parts. This step is crucial in algebra, especially when preparing expressions for further simplification.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They can include addition, subtraction, multiplication, division, and even exponents. Understanding how to manipulate these expressions is key to solving algebra problems effectively.
An algebraic expression like \(7x - 3x(x+1)\) involves multiple operations and variables. It is essential to correctly apply properties, such as the distributive property, to simplify these expressions. Algebraic expressions can represent real-world situations or more abstract mathematical relationships. They often need to be simplified or solved to find an unknown variable's value. Always remember:

• Expressions can include different terms (e.g., \(7x\), \(-3x\)).
• The terms are connected by operators like \(+\), \(-\), and multiplication.
• Applying mathematical properties helps to transform and simplify expressions.

Mastery of recognizing and manipulating algebraic expressions forms the foundation for more advanced algebraic tasks.

Simplifying Expressions

Simplifying expressions is the process of making them easier to work with, typically by combining like terms and arranging them in a conventional order. In mathematics, like terms are terms that contain the same variables raised to the same power, differing only in their coefficients.
In the expression \(7x - 3x^2 - 3x\), the like terms are \(7x\) and \(-3x\).
These terms can be combined by performing the arithmetic operation indicated between them. Here, adding \(7x\) and \(-3x\) results in \(4x\). The expression is simplified further by rearranging it as \(-3x^2 + 4x\) to follow the standard format of listing terms by decreasing powers of the variable.

• Identify and combine like terms to reduce expression complexity.
• Follow conventional ordering for clarity, with strongest terms (highest power) first.

Simplifying expressions not only makes them easier to interpret but also lays the groundwork for solving equations and understanding mathematical relationships.