Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ (5 x+7 y)(3 x+2 y) $$

Step-by-Step Solution

Answer

The expression as a polynomial is \(15x^2 + 31xy + 14y^2\).

Step 1: Distribute the first term in the first binomial

Start by distributing the first term of the first binomial, which is \(5x\), to every term in the second binomial \((3x + 2y)\). Perform the following multiplications: \(5x \times 3x\) and \(5x \times 2y\). This results in: \(15x^2 + 10xy\).

Step 2: Distribute the second term in the first binomial

Now distribute the second term of the first binomial, which is \(7y\), to every term in the second binomial \((3x + 2y)\). Perform the following multiplications: \(7y \times 3x\) and \(7y \times 2y\). This results in: \(21xy + 14y^2\).

Step 3: Combine like terms

Add the results from Step 1 and Step 2 together, combining the like terms \(10xy\) and \(21xy\). Thus, you have: \(15x^2 + 10xy + 21xy + 14y^2\). Combining like terms gives \(15x^2 + 31xy + 14y^2\).

Key Concepts

Binomial Multiplication

Binomial multiplication is a method used to expand expressions involving two polynomials, each with two terms. When multiplying binomials, every term in the first binomial must be multiplied by every term in the second binomial. This technique ensures that none of the possible products are missed.

• Consider the two given binomials: \((5x + 7y)\) and \((3x + 2y)\). This multiplication aims to express the product as a larger polynomial expression.
• The process involves systematically interchanging and multiplying terms from one binomial with those from the other binomial.

Breaking down the expression thoroughly allows the application of further algebraic techniques like the distributive property to organize and simplify the results.

Distributive Property

The distributive property is a crucial algebraic property that aids in multiplying binomials and simplifying expressions. It states that a term multiplied by terms inside a parenthesis must distribute, or apply, to each of those terms.

• In our exercise, we first distribute \(5x\) from the first binomial across the second binomial, producing two products \(5x \times 3x\) and \(5x \times 2y\).
• The same approach is repeated with the second term of the first binomial \(7y\), resulting in \(7y \times 3x\) and \(7y \times 2y\).
• This process forms an intermediary expression with individual terms before any simplification is done: \(15x^2 + 10xy + 21xy + 14y^2\).

Employing the distributive property ensures each term contributes fully to the final expanded expression, maintaining the balance and correctness of algebraic operations.

Combining Like Terms

Combining like terms is an essential step in simplifying polynomial expressions. Like terms are terms that have the same variable raised to the same power.

• In the expanded polynomial: \(15x^2 + 10xy + 21xy + 14y^2\), the terms \(10xy\) and \(21xy\) are like terms because they both contain the same variables \(x\) and \(y\) raised to the same power.
• By adding these together, you combine their coefficients, leading to: \(10xy + 21xy = 31xy\).
• Each term's variable part remains unchanged while you only sum up their coefficients.

The final simplified expression, \(15x^2 + 31xy + 14y^2\), provides a cleaner and more refined format, making it easier to understand and use in further algebraic computations.