Problem 9 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. 2\(x^{2} y\left(y-2 y^{2}\right)\)

Step-by-Step Solution

Answer

The simplest form of the expression is \(2x^2y^2(1 - 2y)\).

Step 1: Distribute the Negative Sign

Begin by distributing the expression inside the parentheses. Note that you are distributing with respect to \(y-2y^2\). This means you should multiply \(y\) by each term. \[y - 2y^2 = y - 2y^2\]

Step 2: Distribute the Outer Term

Multiply \(x^2y\) with each term inside the parentheses to perform the indicated operations.- First, multiply \(x^2y\) with \(y\): \{x^2y imes y = x^2y^2} \- Then, multiply \(x^2y\) with \(-2y^2\): \{x^2y imes -2y^2 = -2x^2y^3} \

Step 3: Combine and Simplify the Expression

After distributing, you have the expression \(2(x^2y^2 - 2x^2y^3)\). Now, factor out the common factor, \(x^2y^2\), from the terms:\[2(x^2y^2 - 2x^2y^3) = 2x^2y^2(1 - 2y)\] This is the expression in its simplest form.

Step 4: Write the Final Result

The simplified form of the given expression after performing the operations is:\[2x^2y^2(1 - 2y)\]

Key Concepts

Distributive Property

When tackling polynomial expressions, the distributive property is essential for simplifying expressions and solving equations. The distributive property allows us to multiply a single term by each term within a parenthesis. For example, in our given expression:

• Multiply the term outside the parentheses, \(x^2y\), by each term inside: \(y - 2y^2\).
• This means we calculate \(x^2y \times y\) and \(x^2y \times -2y^2\).
• The result is \(x^2y^2 - 2x^2y^3\).

Using the distributive property correctly ensures that each term is accurately accounted for, preventing mistakes and maintaining clarity throughout the process.

Simplifying Expressions

Simplifying expressions is a process that reduces them to their simplest form while maintaining their original value. The goal is to make the expression as concise and understandable as possible. After applying the distributive property, we often find ourselves needing to combine like terms or factor out common factors. In this example:

• After distribution, we had the terms \(2(x^2y^2 - 2x^2y^3)\).
• To simplify, notice that \(x^2y^2\) is common in both terms, which means it can be factored out.

This process not only makes expressions easier to work with, but it also highlights patterns or further manipulations that can be applied in solving equations.

Factoring

Factoring is the process of breaking down an expression into simpler components that, when multiplied together, give the original expression. It is a reversing technique from distribution, used here to tidy up the expression:

• In our step, we saw the expression \(2(x^2y^2 - 2x^2y^3)\).
• Factor out the common term \(x^2y^2\) from \(x^2y^2 - 2x^2y^3\).
• The factored form comes out as \(2x^2y^2(1 - 2y)\).

By factoring, we simplify the expression and expose underlying structures, which can be particularly useful in solving polynomial equations or calculus problems.