Problem 70 - College Algebra Essentials

Question

In Exercises 67–82, find each product. $$(3 x-y)(2 x+5 y)$$

Step-by-Step Solution

Answer

The product of \((3x - y)\) and \((2x + 5y)\) gives \(6x^2 + 13xy -5y^2\).

Step 1: Distribute the first terms

Multiply the first terms in each binomial. That will be \(3x * 2x\), which results in \(6x^2\).

Step 2: Distribute the outer terms

Next, Multiply the outer terms in the binomial. That is \(3x * 5y\), resulting in \(15xy\).

Step 3: Distribute the inner terms

Now multiply the inner terms in the binomials. It is \(-y * 2x\), resulting in \(-2xy\).

Step 4: Distribute the last terms

Finally, Multiply the last terms in each binomial. That is, \(-y * 5y\), which results in \(-5y^2\).

Step 5: Combine like terms

Combine the terms from previous steps, it becomes \(6x^2 + 15xy -2xy -5y^2\). Combine similar terms, the \(15xy\) and \(-2xy\), resulting in \(6x^2 + 13xy -5y^2\).

Key Concepts

Polynomial Multiplication

To begin with Polynomial Multiplication, imagine track pieces of a toy train. Each piece connects in different ways to create a longer track. Similarly, polynomials are like these track pieces, and when we multiply them, we assemble them into a bigger polynomial. Polynomials are expressions involving terms like \(3x^2\), \(5xy\), or \(-y^2\). When multiplying polynomials, each term from one polynomial must be multiplied by every term from the other polynomial.

When multiplying polynomials:

• Each term of the first polynomial multiplies each term of the second polynomial.
• The result is a collection of products, which are combined into a polynomial of higher degree.
• Combining like terms simplifies the expression, making it easier to handle.

For example, in the exercise \((3x-y)(2x+5y)\), by multiplying each term from \(3x-y\) with each term in \(2x+5y\), we find the product consisting of similar terms. Finally, simplify these by combining the like terms. This shows how the process structures a new, larger polynomial from smaller parts.

Binomial Expansion

The term Binomial Expansion might sound complex, but it’s just about expanding expressions with two terms, known as binomials. Think of it like stretching a rubber band—each side extends to show what’s hidden inside. In math, binomials such as \(a+b\) or \(3x-y\) are expressions with two parts.

Using the process of Binomial Expansion:

• Apply the distributive property to "stretch" out each term.
• Follow a specific order: First, Outer, Inner, Last—also known as the FOIL method.
• Multiply every term in one binomial with every term in the other.

For instance, in the exercise \((3x-y)(2x+5y)\) we apply FOIL: First— \(3x * 2x\), Outer— \(3x * 5y\), Inner— \(-y * 2x\), and Last— \(-y * 5y\). After expanding, simplify by combining like terms. This expansion method reveals the hidden structure of binomials when combined.

Distributive Property

The Distributive Property is a fundamental part of algebra that allows you to "distribute" multiplication over addition or subtraction within an expression, making complex multiplications simpler. When you distribute, you’re effectively multiplying each term inside a set of parentheses by a term outside it.

To use the Distributive Property:

• Multiply the term outside the parentheses by each term inside.
• Manage both positive and negative signs to ensure accurate computation.

In the context of the given problem \((3x-y)(2x+5y)\), the distributive property is employed in a step-wise manner:

• Multiply the first set of terms \(3x\) by each term \(2x\) and \(5y\).
• Then repeat the process with the second part of the binomial \(-y\).

This property not only simplifies expressions but also serves as a practical tool for solving algebraic equations, as demonstrated by combining the distributive property with Binomial Expansion through FOIL to achieve the desired polynomial result.