Problem 76 - Algebra A Combined Function

Question

Multiply. $$ (y-5 x)(6 y+5 x) $$

Step-by-Step Solution

Answer

The simplified expression is \(6y^2 - 25xy - 25x^2\).

Step 1: Apply Distribution

We start by distributing each term in the first binomial \(y - 5x\) to each term in the second binomial \(6y + 5x\). This follows the distributive property and involves four multiplications: \(y \cdot 6y\), \(y \cdot 5x\), \((-5x) \cdot 6y\), and \((-5x) \cdot 5x\).

Step 2: Simplify Each Product

Calculate each of the products from the distribution: 1. \(y \cdot 6y = 6y^2\)2. \(y \cdot 5x = 5xy\)3. \((-5x) \cdot 6y = -30xy\)4. \((-5x) \cdot 5x = -25x^2\)

Step 3: Combine Like Terms

Combine the like terms from the simplified products. Specifically, combine \(5xy\) and \(-30xy\) to get \(-25xy\). So, the entire expression becomes: \[ 6y^2 + 5xy - 30xy - 25x^2 \]which simplifies to \[ 6y^2 - 25xy - 25x^2 \].

Key Concepts

Distribution Property

The distribution property is a fundamental principle in algebra that allows us to multiply a single term across the terms within a bracket. This is particularly useful when working with polynomials, especially when multiplying two binomials. In the original exercise, we have two binomials:

• \(y - 5x\)
• \(6y + 5x\)

To apply the distribution property, you need to systematically multiply each term in the first binomial by each term in the second binomial. This results in four separate products:

• \(y \cdot 6y\)
• \(y \cdot 5x\)
• \((-5x) \cdot 6y\)
• \((-5x) \cdot 5x\)

When approached step by step, the distribution property is like "unpacking" the layers of expression, allowing us to handle and simplify each multiplication before moving on to complete the entire expression.

Simplifying Expressions

Simplifying expressions refers to reducing an expression to its most concise form. After distributing, you'll end up with several products that need to be simplified. In mathematical expressions, simplification often involves combining like terms, factoring, or canceling out terms where possible.First, simplify each product individually:

• \(y \cdot 6y = 6y^2\)
• \(y \cdot 5x = 5xy\)
• \((-5x) \cdot 6y = -30xy\)
• \((-5x) \cdot 5x = -25x^2\)

These results allow us to see the expanded polynomial more clearly:\[ 6y^2 + 5xy - 30xy - 25x^2 \]Each term now stands independently, ready to be further simplified by combining like terms, a concept we will discuss next.

Combining Like Terms

Combining like terms is a method used to simplify an algebraic expression by combining terms that have the exact same variable parts. This is crucial in organizing a polynomial into its simplest form after applying the distribution property and simplifying each term.Upon distributing and simplifying, we have:\[ 6y^2 + 5xy - 30xy - 25x^2 \]Notice that the terms \(5xy\) and \(-30xy\) have identical variable parts, \(xy\). This allows us to combine them:- \(5xy - 30xy = -25xy\)The expression then becomes:\[ 6y^2 - 25xy - 25x^2 \]This consolidation of like terms makes the polynomial much easier to understand and utilize in further calculations. Look for these opportunities to combine terms every time you are simplifying an expression.