Problem 70 - Algebra 2

Question

PREREQUISITE SKILL. Use the Distributive Property to find each product. $$ -2(3 x-5) $$

Step-by-Step Solution

Answer

The product is \(-6x + 10\).

Step 1: Identify the Expression

We are given the expression \[ -2(3x - 5) \] which consists of \(-2\) multiplied by the binomial \((3x - 5)\). Our task is to distribute the \(-2\) over the terms within the parentheses.

Step 2: Apply the Distributive Property

The Distributive Property states that \(a(b + c) = ab + ac\). In this problem, substitute \(a = -2\), \(b = 3x\), and \(c = -5\). Distribute \(-2\) across \(3x\) and \(-5\) as follows:\[ \begin{align*} -2(3x) & \quad+\quad -2(-5) \=-6x & \quad+\quad 10 \end{align*} \]

Step 3: Simplify Each Term

Now that we've distributed, simplify each term:- From \(-2 \times 3x\), we have \(-6x\).- From \(-2 \times -5\), we obtain \(+10\).Combine these simplified terms for the final expression:\[ -6x + 10 \]

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators, like addition, subtraction, multiplication, and division, that form a meaningful representation in mathematics. These expressions can range from simple to complex, based on the number of variables and terms they contain.
In our exercise, the expression \( -2(3x - 5) \) is considered an algebraic expression because it includes a variable \(x\), along with numbers and the multiplication operation. The task is to manipulate this expression using mathematical properties to simplify it.
Different operators are used to perform operations on these algebraic expressions:

• **Addition and subtraction** to change or reduce the value of the expression.
• **Multiplication and division** to distribute numbers across terms or simplify ratios.

When working with algebraic expressions, understanding how to handle variables and constants, and how different operations affect them, is essential.

Binomials

A binomial is a type of algebraic expression that has exactly two terms. These terms are separated by either a plus or a minus sign. For example, in the expression \(3x - 5\), \(3x\) and \(-5\) are the two terms of the binomial.
Recognizing binomials is crucial because they often require specific strategies when performing operations like addition, subtraction, or multiplication. In our exercise, we specifically looked at multiplying a binomial by a constant using the Distributive Property.
One should remember these characteristics of binomials:

• Two distinct terms.
• Terms typically involve a combination of variables and constants.
• The sign between the terms dictates the operation (either addition or subtraction).

Multiplying or expanding binomials often involves distributing an exterior multiplier across each term within the binomial.

Simplification

Simplifying algebraic expressions is the process of reducing them to their simplest form. This involves combining like terms, performing arithmetic operations, and using mathematical properties like the Distributive Property.
In our given problem, after using the Distributive Property on the expression \( -2(3x - 5) \), we expanded it to \(-6x + 10\). The expression is simplified by ensuring like terms are combined correctly and ensuring each term is at its most reduced form.
Simplification helps in:

• Making expressions more manageable for further calculations or solving equations.
• Highlighting key structural elements that may not be obvious in a more complex form.

Efficient simplification demands not only calculations but also a solid understanding of mathematical properties and how they apply to various algebraic structures.