Problem 69 - Algebra and Trigonometry

Question

In Exercises 67–82, find each product. $$ (x-3 y)(2 x+7 y) $$

Step-by-Step Solution

Answer

The product of \((x-3y)(2x+7y)\) is \(2x^2+xy-21y^2\).

Step 1: Multiplying the First Terms

Multiply the first terms in each binomial. This means it's necessary to multiply \(x\) and \(2x\) together. The product of these two terms is \(2x^2\).

Step 2: Multiplying the Outer Terms

Multiply the outer terms in the expression, which means we have to multiply the \(x\) in the first binomial by \(7y\) in the second binomial, giving us \(7xy\).

Step 3: Multiplying the Inner Terms

Multiply the inner terms. The two inner terms here are \(-3y\) and \(2x\), which when multiplied together give \(-6xy\).

Step 4: Multiplying the Last Terms

Multiply the last terms in each binomial together. This involves multiplying \(-3y\) and \(7y\) together, yielding \(-21y^2\).

Step 5: Summing Up All Products

Add together all of the products obtained from Steps 1-4. The expression becomes \(2x^2+7xy-6xy-21y^2\). By simplifying the like terms, the result yields \(2x^2+xy-21y^2\).

Key Concepts

Understanding Binomials

Binomials are simple forms of algebraic expressions that contain exactly two terms. These terms are typically separated by either a plus or a minus sign. For example, the expression \(x - 3y\) is a binomial with two distinct terms: \(x\) and \(-3y\).
Binomials can often be encountered in various algebraic expressions and equations, making them extremely important in solving polynomial equations. They act as building blocks for more complex expressions.
When dealing with binomials, it’s crucial to recognize their structure so we can apply the appropriate arithmetic operations, such as addition, subtraction, and especially multiplication, which we'll focus on here. Understanding binomials lays the groundwork for mastering polynomial arithmetic.

The Distributive Property in Polynomial Multiplication

The distributive property is a fundamental concept in algebra that allows us to simplify the process of multiplying polynomials, including binomials. This property states that for any numbers or algebraic expressions \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true.
This principle is crucial in polynomial multiplication. It ensures that we consider each term in one binomial with each term in the other. In the exercise provided, we must multiply \((x - 3y)\) by \((2x + 7y)\).
The distributive property helps us organize this into parts: first, outer, inner, and last terms multiplication:

• First: Multiply the first terms from each binomial \(x\) and \(2x\), resulting in \(2x^2\).
• Outer: Multiply the outer terms \(x\) and \(7y\), giving \(7xy\).
• Inner: Multiply the inner terms \(-3y\) and \(2x\), resulting in \(-6xy\).
• Last: Finally, multiply the last terms \(-3y\) and \(7y\), which yields \(-21y^2\).

After performing these steps, all products are combined and simplified according to the rules of algebra.

Deeper Dive into Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can range from simple terms, like \(2x\), to more complex ones, such as \(2x^2 + xy - 21y^2\). Each part of an expression has a specific role and purpose.
The expressions that we frequently work with in algebra are often polynomial expressions, which are sums of several terms.
In the exercise, after applying the distributive property, we end up with the expression \(2x^2 + 7xy - 6xy - 21y^2\). To simplify this expression, we must combine like terms, which refers to terms that have exactly the same variables raised to the same power.
For instance, the \(7xy\) and \(-6xy\) terms are like terms. By combining them through addition or subtraction as appropriate, we simplify the expression to \(2x^2 + xy - 21y^2\).
Understanding algebraic expressions, and their simplification, is vital as it makes solving complex equations and polynomials more accessible and less daunting.