Problem 13 - Elementary Algebra

Question

Multiply. $$ -5(3 x-2) $$

Step-by-Step Solution

Answer

The result is \(-15x + 10\).

Step 1: Distribute the Constant

Begin the multiplication by distributing the constant \(-5\) to each term inside the parenthesis. This means multiplying \(-5\) by \(3x\) and \(-5\) by \(-2\).

Step 2: Multiply \\(-5\\) by the Variable Term

Calculate \(-5\) times \(3x\):\[-5 \times 3x = -15x\].

Step 3: Multiply \\(-5\\) by the Constant Term

Calculate \(-5\) times \(-2\):\[-5 \times (-2) = 10\].

Step 4: Combine the Results

Combine the results from Step 2 and Step 3 to form the final expression: \(-15x + 10\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are fundamental components in algebra that consist of constants, variables, and operations. At their core, they are combinations of numbers and letters that represent quantities and relationships. For instance, in the expression \(3x - 2\), the number 3 is a coefficient, \(x\) is a variable, and -2 is a constant term.
When working with algebraic expressions, it's important to understand their structure. Variables can take on different values, which means that the expression as a whole can change.

• Coefficients are the numbers that multiply the variables.
• Constant terms are standalone numbers without variables.

Recognizing these parts helps in operations like addition, subtraction, and multiplication, leading to the simplification or expansion of expressions. In our example, distributing the -5 across each term in the expression \(3x - 2\) involves both the variable and the constant.

Handling Negative Numbers

Negative numbers may seem confusing at first, but they follow specific rules that can simplify calculations. When you multiply a negative number by a positive, the result is always negative. However, when two negative numbers are multiplied, they yield a positive result.
In the exercise, you multiply two negative numbers in Step 3: \(-5 \times (-2)\), which results in 10. Understanding the rules of negative numbers is crucial:

• A positive times a negative yields a negative.
• A negative times a negative yields a positive.

Keeping these principles in mind can help students approach problems with confidence, ensuring accurate solutions.

Multiplication of Polynomials

The multiplication of polynomials is a key operation in algebra. It involves distributing each term in one polynomial to each term in another. This is often done using the distributive property, which ensures each term in the polynomials is properly accounted for.
In our example, distributing the \(-5\) to \(3x-2\) involves:

• Multiplying \(-5\) by the first term \(3x\), which gives \(-15x\).
• Next, multiplying \(-5\) by the constant \(-2\), resulting in 10.

The final expression \(-15x + 10\) is a simplified polynomial that combines these results. Understanding this process involves recognizing each term in the polynomial and applying multiplication rules effectively. This ensures that the expression is simplified correctly, and all terms are accounted for.