Problem 26 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \\(-6x^3 + 4x^2 - 2x + 6\\).

Step 1: Distribute the -2

To multiply the expression, you start by distributing the -2 to each term inside the parentheses. This means you multiply -2 by each of the four terms separately: 1. \(-2 imes 3x^3\)2. \(-2 imes (-2x^2)\)3. \(-2 imes x\)4. \(-2 imes (-3)\).

Step 2: Multiply -2 by Each Term

Perform the multiplication for each term:1. \(-2 imes 3x^3 = -6x^3\)2. \(-2 imes (-2x^2) = 4x^2\)3. \(-2 imes x = -2x\)4. \(-2 imes (-3) = 6\).

Step 3: Combine the Results

Now, combine the results from Step 2 to write the final, simplified expression:\[-6x^3 + 4x^2 - 2x + 6\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions like the one in our exercise. It involves distributed multiplication over addition or subtraction inside parentheses. Imagine you have an expression like \(a \cdot (b + c)\)\. The distributive property tells us that this is the same as \(a \cdot b + a \cdot c\)\. This property holds true for subtraction as well, meaning \(a \cdot (b - c)\) becomes \(a \cdot b - a \cdot c\)\.
In our exercise, we have \(-2(3x^3 - 2x^2 + x - 3)\)\. Here, we apply the distributive property by multiplying \(-2\) with each term inside the parentheses, one at a time. This step ensures each term is affected by the \(-2\) multiplication, which sets the stage for combining and simplifying the expression in later steps.

Negative Multiplication

When dealing with negative multiplication, the rules can sometimes seem tricky, but they are straightforward once you understand them. Multiplying a negative number by a positive number flips the sign of the product. For example, multiplying \(-2 \cdot 3x^3\) results in \(-6x^3\)\. What's crucial is to keep track of the signs to ensure accuracy.
Here's where it gets more interesting: multiplying a negative number by another negative number results in a positive product. This is why in our problem, when we multiply \(-2 \cdot (-2x^2)\), we end up with a positive \(4x^2\)\. In a similar way, \(-2 \cdot (-3) = 6\) produces a positive result. Remembering these basic rules about signs while multiplying will help in tackling even more complex expressions.

Simplifying Expressions

Simplifying expressions involves more than just performing arithmetical operations; it's about expressing the result in its most concise form. Once you have distributed and completed multiplications, like in our example, you'll want to combine all terms into a clean, coherent expression.
From the example, we're left with the terms \(-6x^3\), \(4x^2\), \(-2x\), and \(6\). Each term represents a part of the polynomial, with different powers of \(x\) and a constant term. Since no terms are similar in degree or type, we cannot further combine them but simply write them out in descending order of degree. This gives us our final expression: \(-6x^3 + 4x^2 - 2x + 6\).
By fully simplifying, we ensure that the expression is easy to read and understand. It also helps in identifying key features of the polynomial, like its degree and leading coefficient, important in further mathematical analysis or application.