Problem 28 - Algebra 2

Question

Simplify. $$ 2 x y\left(3 x y^{3}-4 x y+2 y^{4}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(6x^2y^4 - 8x^2y^2 + 4xy^5\).

Step 1: Distribute the Terms

To simplify the expression, distribute the term outside the parenthesis, \(2xy\), to each term inside the parenthesis, \(3xy^3 - 4xy + 2y^4\). This yields:1. \(2xy \cdot 3xy^3 = 6x^2y^4\)2. \(2xy \cdot -4xy = -8x^2y^2\)3. \(2xy \cdot 2y^4 = 4xy^5\)

Step 2: Write the Simplified Expression

Combine the distributed terms into a single expression to complete the simplification. The expression now becomes:\[6x^2y^4 - 8x^2y^2 + 4xy^5\]

Key Concepts

Distributive Property

The distributive property is a fundamental tool in algebra, particularly useful for simplifying expressions and equations. It allows us to multiply a term outside a parenthesis by each term inside, spreading the multiplication across the addition or subtraction inside. For the expression given, we apply the distributive property by taking the term outside, which is \(2xy\), and multiplying it with each term inside the parenthesis: \(3xy^3\), \(-4xy\), and \(2y^4\).

• For \(2xy \cdot 3xy^3\), you multiply the numerical coefficients (2 and 3) to get 6, then apply multiplication of powers for variables \(x\) and \(y\). This results in \(6x^2y^4\).
• Next, \(2xy \cdot -4xy\) gives us \(-8x^2y^2\), following similar steps with coefficients and powers.
• Finally, \(2xy \cdot 2y^4\) results in \(4xy^5\), keeping the \(x\)'s power unchanged while adding the powers of \(y\).

The distributive property is crucial when managing expressions with multiple terms and variables, simplifying into manageable equations.

Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients, structured into terms connected by addition or subtraction. Each term typically includes a variable raised to a power (the exponent), multiplied by a numerical coefficient. Understanding polynomials involves recognizing these components and knowing how to manipulate them.

• In our example, terms like \(3xy^3\), \(-4xy\), and \(2y^4\) form a polynomial inside the parenthesis. Each term has its own set of coefficients and variables.
• A key aspect of working with polynomials is the ability to reorganize and combine like terms to simplify the expression further, which is often necessary after distributing multiplications.
• Polynomials of different degrees involve different powers of variables; here, powers of \(y\) vary from 2 to 5, showing the polynomial nature.

Handling polynomials efficiently requires recognizing patterns and using algebraic rules, ensuring each term is appropriately managed.

Multiplication of Variables

Multiplying variables in algebra follows straightforward rules, especially when dealing with powers and exponents. When you multiply variables, each with an exponent, you add the exponents of like bases, following the law \(a^m \cdot a^n = a^{m+n}\). This principle simplifies what might initially seem complicated.

• For instance, in \(2xy \cdot 3xy^3\), you multiply the coefficients (2 and 3), then apply the law of exponents: the \(x\)'s exponents \((1+1)\) yield \(x^2\), while \(y\)'s \((1+3)\) yield \(y^4\).
• This step-by-step approach ensures each variable is correctly accounted for, without altering the overall operation improperly.

Understanding multiplication of variables not only aids in simplifying expressions but also instills deeper comprehension of algebra's foundations, empowering further exploration and manipulation of algebraic expressions.