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Chapter 6: Problem 4

Use the distributive property to expand each expression. $$ \frac{1}{2} x(x-2) $$

Short Answer

Expert verified
\(\frac{1}{2}x^2 - x\)

Step by step solution

01

Identify the Expression

The given expression is \(\frac{1}{2}x(x-2)\).
02

Apply the Distributive Property

Use the distributive property, \(a(b+c)=ab+ac\), to distribute \(\frac{1}{2}x\) to both \(x\) and \(-2\).
03

Multiply \(\frac{1}{2}x\) by \(x\)

Calculate \(\frac{1}{2}x \times x\) to get \(\frac{1}{2}x^2\).
04

Multiply \(\frac{1}{2}x\) by \(-2\)

Calculate \(\frac{1}{2}x \times -2\) to get \(-x\).
05

Combine the Results

Combine the multiplied terms to get the expanded expression: \(\frac{1}{2}x^2 - x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expanding Expressions
Expanding expressions involves using the distributive property to simplify algebraic expressions. The goal is to remove any parentheses by distributing the multiplication over addition or subtraction within them. Suppose you have an expression like \(\frac{1}{2}x(x-2)\). To remove the parentheses, you distribute \(\frac{1}{2}x\) to each term inside the parenthesis. This step ensures each part of the expression is multiplied properly.
Multiplying Variables
When multiplying variables, it鈥檚 important to follow the basic rules of algebra. Let's look at the expression \(\frac{1}{2}x \times x\). Here, you apply the multiplication rule for powers: \(a^m \times a^n = a^{m+n}\). Therefore, \(\frac{1}{2}x \times x\) becomes \(\frac{1}{2}x^{1+1} = \frac{1}{2}x^2\). This is because multiplying x by x raises its exponent by 1.
Combining Like Terms
Combining like terms simplifies expressions by merging terms with the same variable and exponent. For instance, in the expanded form \(\frac{1}{2}x^2 - x\), we identify terms with the same variable. While \(\frac{1}{2}x^2\) and \-x\ are already simplified, you ensure that coefficients are also simple. This process helps to streamline algebraic expressions, making them easier to understand and solve.

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Most popular questions from this chapter

Simplify each fraction. $$ \frac{15}{75} $$

Expand each expression. Simplify your expansion if possible. \((4 x+1)^{2}\)

Use the distributive property to rewrite each expression without parentheses. $$ -(3 n-4) $$

Expand and simplify each expression. $$ (x+2.5)^{2} $$

Find each product or quotient. Simplify your answers. $$ \frac{m}{4} \div \frac{4}{m} $$
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