Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ \left(x^{2}+2 y\right)\left(x^{2}-2 y\right) $$

Step-by-Step Solution

Answer

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

Step 1: Recognize the Expression Form

The expression \((x^2 + 2y)(x^2 - 2y)\) is in the form \((a + b)(a - b)\). This is the difference of squares pattern, which can be expanded using the identity \((a + b)(a - b) = a^2 - b^2\). Here, let \(a = x^2\) and \(b = 2y\).

Step 2: Apply the Difference of Squares Formula

Using the identity from Step 1, apply \((x^2)^2 - (2y)^2\). Substitute back for \(a\) and \(b\):\[a^2 = (x^2)^2 = x^4\]\[b^2 = (2y)^2 = 4y^2\]The polynomial becomes:\[x^4 - 4y^2\]

Step 3: Simplify the Polynomial

The expression \(x^4 - 4y^2\) is already in its simplest polynomial form, as no further factorization or terms are required.

Key Concepts

Difference of Squares

The expression you see, \((x^2 + 2y)(x^2 - 2y)\), is a classic example of the difference of squares. This principle states that any binomial expression of the form \((a + b)(a - b)\) can be expanded into \(a^2 - b^2\). A key feature of the difference of squares is that it always involves subtracting one square from another.
For our expression, we identify \(a = x^2\) and \(b = 2y\). Thus, the formula \(a^2 - b^2\) becomes \((x^2)^2 - (2y)^2\). This simplifies to \(x^4 - 4y^2\).
This elegant result shows how such seemingly complex expressions can be transformed simply by recognizing the difference of squares pattern. It is quite useful in algebra as it makes problems easier to simplify or solve.

Algebraic Expressions

Algebraic expressions are made up of variables, constants, and operation signs like addition, subtraction, multiplication, and division. In the expression \((x^2 + 2y)(x^2 - 2y)\), the terms inside the parentheses, \(x^2 + 2y\) and \(x^2 - 2y\), are called binomials. A binomial is a specific type of algebraic expression with exactly two terms.
Variables like \(x\) and \(y\) in algebraic expressions can stand for numbers and are quite flexible—they can change in value. Constants, like the 2 in \(2y\), are fixed values. The objective here is to utilize arithmetic operations to manipulate these expressions further, often simplifying them or solving for specific variables.

Polynomial Expansion

Polynomial expansion involves expanding products of expressions to write them as a sum of terms. For instance, expanding \((x^2 + 2y)(x^2 - 2y)\) involves applying the difference of squares pattern.
Using the formula, \((a + b)(a - b) = a^2 - b^2\), the polynomial gets expanded to \(x^4 - 4y^2\). This converts the original product into a single expression.
The result of expanding a polynomial is a new polynomial of terms, where each term is the product of two or more terms from the original expression. Here, our expression simplifies directly to \(x^4 - 4y^2\), consisting of fewer terms and making it straightforward to work with in algebra.