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Chapter 1: Problem 50

Find each product or quotient. $$ x(-x) $$

Short Answer

Expert verified
-x^2

Step by step solution

01

- Understand the Expression

The given expression is the product of two terms: \(x \times (-x)\).
02

- Apply the Multiplication Rule

Recall that multiplying a variable by its negative is the same as multiplying the numerical coefficient: \(a \times (-a) = -a^2\).
03

- Multiply the Terms

Now, multiply the terms: \(x \times (-x) = -x^2\).

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Key Concepts

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

Negative Multiplication
Multiplying negative numbers can seem tricky, but it's straightforward if you follow some simple rules. When you multiply a positive number by a negative number, the product will always be negative. Imagine you have a variable, say, **x**.
If you multiply it by its negative, like \((-x)\), the result is similar to multiplying a positive and a negative number. So, \(x \times (-x)\) will always give you \(-x^2\).
  • Positive \( \times \) Negative = Negative
  • Negative \( \times \) Positive = Negative
This concept is fundamental in algebra and essential for solving problems involving negative numbers and variables.
Algebraic Expressions
Algebraic expressions are mathematical phrases that include numbers, variables, and operators (like addition, subtraction, multiplication, and division). For example, in the expression **x(-x)**:
  • **x** is a variable, a symbol that represents a number we often don't know yet.
  • **-x** is the negative of the variable, signifying that whatever value **x** holds, **-x** will be its opposite in sign.
  • **x(-x)** is an algebraic expression where you are multiplying two variables 鈥� one positive and one negative.
  • When you multiply variables, you follow the same multiplication rules as you do with numbers, adjusting for their signs.
Algebraic expressions are the building blocks of algebra and necessary for understanding and solving equations.
Product of Terms
When dealing with the multiplication of terms, understanding each part of the expression is crucial. In our case, we are multiplying **x** by **-x**. Here's a step-by-step breakdown:
  • Identify the terms: The terms here are **x** and **-x**.
  • Apply product rules: The product of variables is the product of their coefficients (numbers in front of the variables) and then the product of the variables themselves.
  • For **x** and **-x**, the coefficient is 1 for both, but one is positive and the other is negative.
  • Multiply the coefficients: \(1 \times -1 = -1\).
  • Multiply the variables: \(x \times x = x^2\).
  • Combine them to get the final product: \-x^2\.
The product of terms when multiplied under these rules ensures accuracy and clarity in solving algebraic problems.

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