Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The polynomial is \(4x^2 - 9y^2\).

Step 1: Identify the Formula

The expression \((2x + 3y)(2x - 3y)\) is a product of two binomials. This fits the structure of the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\). Here, \(a = 2x\) and \(b = 3y\).

Step 2: Apply the Difference of Squares Formula

Substitute the identified values into the difference of squares formula: \((2x + 3y)(2x - 3y) = (2x)^2 - (3y)^2\).

Step 3: Square Each Term

Now, compute \((2x)^2\) which gives \(4x^2\) and \((3y)^2\) which gives \(9y^2\).

Step 4: Write the Resulting Polynomial

Substitute the squared terms back into the expression: \(4x^2 - 9y^2\). This is the polynomial expression of the original product.

Key Concepts

Binomial Product

A binomial product arises when two binomials are multiplied together.
In this context, a binomial is an expression containing two terms, usually joined by a plus or minus sign.
For example, in \((2x + 3y)(2x - 3y)\), each part inside the brackets is a binomial.

• "2x + 3y" is one binomial.
• "2x - 3y" is the second binomial.

The operation involves distributing each term in the first binomial to each term in the second binomial.
This requires us to perform multiplication for each pair of terms, add-like terms, and finally simplify the result.
The process usually follows the principle of FOIL (First, Outer, Inner, Last). However, when the binomials are in the form \((a+b)(a-b)\), the product can be simplified using a special product called the difference of squares.
This makes the calculation much faster and cleaner.

Difference of Squares

The difference of squares is a specific algebraic pattern that speeds up multiplication of specific binomials.
The formula is given as \((a+b)(a-b) = a^2 - b^2\).
This tells us that the product of binomials \((a+b)\) and \((a-b)\) will yield the difference between their squares. Understanding the components:

• "a" and "b" are any algebraic expressions.
• "a²" and "b²" are simply a and b each squared.

In the given exercise, \((2x + 3y)(2x - 3y)\), we identify "a" with "2x"and "b" with "3y".
Applying the formula results in \((2x)^2 - (3y)^2\), which simplifies to \(4x^2 - 9y^2\).
This technique is faster than using individual distribution because it skips the step of dealing with middle terms, summarizing the result in just two parts; the squares of each term.

Polynomial Expansion

Polynomial expansion is the process of simplifying an expression with multiple terms.
Each binomial multiplication produces a set of terms that can often be expanded into a simpler or more manageable form.
By expanding, we translate the multiplication directly into a straightforward expression of sums and powers without parentheses.Key steps in polynomial expansion include:

• Identifying the terms to multiply.
• Using basic algebraic principles to distribute terms.
• Simplifying each part to its basic polynomial form.

When applying the difference of squares, the completed expansion for \((2x + 3y)(2x - 3y)\) results in the polynomial\(4x^2 - 9y^2\).
This polynomial does not leave any terms undistributed or without coefficients, making it easy to read and use in further calculations.