Problem 28 - Elementary and Intermediate Algebra

Question

Multiply. See Example 2. $$ -6 s\left(s^{2}-3\right) $$

Step-by-Step Solution

Answer

The product is \(-6s^{3} + 18s\).

Step 1: Identify the Terms

The expression you need to multiply is \(-6s(s^2 - 3)\). Identify the terms inside and outside the parentheses. Here, we have the number and variable \(-6s\) outside the parentheses and \(s^2 - 3\) inside the parentheses.

Step 2: Apply the Distributive Property

To multiply the expression, use the distributive property, which states that \(a(b + c) = ab + ac\). Here, \(a = -6s\), \(b = s^2\), and \(c = -3\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra. It allows us to simplify expressions by distributing a multiplier across terms inside a set of parentheses. Put simply, for any numbers or expressions, if you have something like \(a(b + c)\), the distributive property tells you that it equals \(ab + ac\). This property helps us expand expressions and solve equations more easily.

In the given problem, we have to expand the expression \(-6s(s^2 - 3)\). Using the distributive property means taking \(-6s\) and multiplying it with each term inside the parentheses separately. First, \(-6s\) is multiplied with \(s^2\), and then with \(-3\). This step-by-step distribution makes complex equations simple and manageable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. These expressions represent various quantities and relationships in algebra. Understanding how to manipulate algebraic expressions is essential for problem-solving in mathematics.

In our exercise, the expression \(-6s(s^2 - 3)\) is an example of an algebraic expression needing multiplication to simplify it. The goal is to break down or rearrange these expressions according to specific rules, like the distributive property, allowing us to arrive at a clearer or more solved form of the expression. When working with expressions:

• Identify any constants, coefficients, and variables
• Understand the operations to perform on them
• Apply algebraic principles such as the distributive property to rearrange or simplify

Mastery of these expressions helps in solving a wide range of mathematical problems.

Intermediate Algebra

Intermediate algebra builds on the basics of elementary algebra, diving deeper into functions, polynomial equations, and complex numbers. A common task in intermediate algebra is polynomial multiplication, similar to what is seen in our problem \(-6s(s^2 - 3)\). It demands a solid grasp of both basic algebraic rules and more advanced techniques.

When dealing with intermediate algebra problems:

• Ensure a strong foundation in basic operations and properties, such as the distributive property
• Practice manipulating algebraic expressions until the process is intuitive
• Evaluate more complex structures, which may include quadratic and cubic terms
• Apply strategies systematically to simplify or expand these expressions

By mastering these concepts, solving complex mathematical problems becomes more approachable, paving the way for advanced studies in mathematics and related fields.