Problem 44 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -2 x(x-8) $$

Step-by-Step Solution

Answer

The expression -2x(x-8) without parentheses, using the distributive property, is \(-2x^2 + 16x\).

Step 1: Understanding the Problem

Here we're asked to use the distributive property to rewrite the expression without parentheses. The distributive property is \( -2x * x - (-2x * 8) \). This means that each term inside of the parentheses needs to be multiplied by the term outside.

Step 2: Distribute the -2x to Each Term Inside the Parentheses

Apply the distributive property: \( -2x * x + (-2x * -8) \). This results in two separate terms: \(-2x^2 + 16x \).

Step 3: Checking the Answer

Now that the expression is without parentheses, check the work by using the distributive property backward. Multiply -2x by each term in \( -2x^2 + 16x \) and confirm it results in the original expression \( -2x(x-8) \). If it does, then the work is complete.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as addition, subtraction, multiplication, and division). For example, in the expression -2x(x-8), -2x and (x-8) are two algebraic expressions being multiplied together. The variable x represents an unknown value, and the -2 in front of it is a constant that indicates the variable is being multiplied by -2.

Algebraic expressions are fundamental in forming algebraic equations, which can help describe relationships and solve problems involving unknown quantities. Understanding how to manipulate these expressions correctly using various properties of operations is essential in simplifying them and solving algebraic problems.

Simplifying Expressions

Simplifying expressions means to reduce an algebraic expression to its simplest form. This involves combining like terms, which are terms that have the same variables raised to the same power, and carrying out any operations according to the order of operations ('PEMDAS'—Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

For instance, in the expression -2x(x-8), by applying the distributive property, we can simplify the expression to -2x^2 + 16x. This result no longer has parentheses, making it a simplified version of the original expression. It is crucial to understand that simplifying an expression does not change its value; it merely presents it in a more straightforward and digestible manner for further mathematical operations or problem-solving.

Properties of Operations

Properties of operations are the rules that apply to different mathematical operations, which help us manipulate and simplify expressions. One key property is the distributive property, which allows us to multiply a single term across the terms within a set of parentheses.

For example, using the distributive property on the expression -2x(x-8), we multiply the -2x by each term inside the parentheses. This gives us -2x^2 when we multiply -2x by x, and +16x when we multiply -2x by -8 (remember, a negative times a negative equals a positive). Understanding and applying the distributive property correctly is crucial in solving algebraic expressions, as it helps in breaking down more complex expressions into simpler components that are easier to handle.