Problem 63 - Elementary Algebra

Question

Simplify. $$ -3 x(x-1)+5 x(x 2-1) $$

Step-by-Step Solution

Answer

The simplified expression is \(5x^3 - 3x^2 - 2x\).

Step 1: Expand the Expression

First, we'll distribute the terms inside both parentheses:For \(-3x(x - 1)\):\(-3x \cdot x + (-3x) \cdot (-1) = -3x^2 + 3x\)For \(5x(x^2 - 1)\):\(5x \cdot x^2 + 5x \cdot (-1) = 5x^3 - 5x\)

Step 2: Combine Like Terms

Write down all the expanded terms and combine the like terms:\[-3x^2 + 3x + 5x^3 - 5x\]Group by similar powers of \(x\):\[5x^3 - 3x^2 + 3x - 5x\]Combine the linear terms:\[5x^3 - 3x^2 - 2x\]

Step 3: Write the Final Simplified Expression

The simplified expression, after combining all like terms, is:\(5x^3 - 3x^2 - 2x\)

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. These expressions are fundamental in algebra because they allow us to represent and manipulate quantities in a flexible way. For example, the expression \[-3x(x - 1) + 5x(x^2 - 1)\]is an algebraic expression that contains variables \(x\), coefficients like \(-3\) and \(5\), and operations like multiplication and subtraction.

Here are some key points about algebraic expressions:

• Variables: Symbols, often letters, that represent numbers. In our expression, \(x\) is a variable.
• Coefficients: Numbers placed in front of variables to indicate multiplication. For instance, in \(-3x\), \(-3\) is the coefficient.
• Operators: Symbols that show mathematical operations. In this case, we have addition, subtraction, and multiplication.

Understanding how to manipulate these components is essential for simplifying expressions.

Distributive Property

The distributive property is a useful algebraic property that allows us to multiply a single term by more than one term within parentheses. This property is especially handy when simplifying expressions like the one in our exercise.

In the expression \[-3x(x - 1) + 5x(x^2 - 1)\],we utilize the distributive property to simplify it:

• For \(-3x(x - 1)\), we distribute \(-3x\) to each term within the parentheses, resulting in \(-3x \cdot x + (-3x) \cdot (-1) = -3x^2 + 3x\).
• Similarly, for \(5x(x^2 - 1)\), apply the distributive property to get \(5x \cdot x^2 + 5x \cdot (-1) = 5x^3 - 5x\).

The distributive property helps us break down complex expressions into simpler parts, making them easier to work with and combine further. Once distributed, each polynomial piece is easier to handle, facilitating the next step in simplification.

Combining Like Terms

After using the distributive property, the next step in simplifying algebraic expressions involves combining like terms. Like terms are terms that contain the same variables raised to the same power. Only the coefficients of these terms are different.

In our simplified components \[-3x^2 + 3x + 5x^3 - 5x\],we follow these steps:

• Identify and group the terms with the same power of \(x\): Here, we have \(5x^3\), \(-3x^2\), and the linear terms \(3x\) and \(-5x\).
• Combine the linear terms by adding their coefficients: \(3x - 5x = -2x\).

After combining like terms, the expression becomes \[5x^3 - 3x^2 - 2x\].This method simplifies the expression to its final form, making it more straightforward to understand and solve further if needed. Combining like terms is a crucial algebraic skill, particularly when dealing with polynomials.