Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The expression simplifies to the polynomial \(x^2 - 2xy + y^2\).

Step 1: Recognize the Formula

The expression \((\sqrt{x}+\sqrt{y})^{2}(\sqrt{x}-\sqrt{y})^{2}\) is a product of two squares. We can employ the formula for the product of squares of the difference of squares: \((a+b)^2(a-b)^2 = ((a+b)(a-b))^2 = (a^2 - b^2)^2\).

Step 2: Apply the Formula

Let \(a = \sqrt{x}\) and \(b = \sqrt{y}\). Apply the formula: \(((\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y}))^2\). The intermediate expression becomes \(((\sqrt{x})^2 - (\sqrt{y})^2)^2\).

Step 3: Simplify the Inner Expression

Simplify \(((\sqrt{x})^2 - (\sqrt{y})^2)\) to \((x - y)\). This step simplifies our expression to \((x - y)^2\).

Step 4: Expand the Polynomial

Use the formula for the square of a binomial to expand \((x-y)^2\). It becomes \(x^2 - 2xy + y^2\).

Step 5: Write the Polynomial

Now, we express \((\sqrt{x}+\sqrt{y})^{2}(\sqrt{x}-\sqrt{y})^{2}\) as the polynomial \(x^2 - 2xy + y^2\).

Key Concepts

Product of Squares

In algebra, the concept of the "Product of Squares" is often used to transform expressions into simpler or more useful forms. Consider the expression \((a+b)^2(a-b)^2\). This represents the product of two squares: the square of \((a+b)\) and the square of \((a-b)\). The beauty of this form is its transformation through the identity:

• \((a+b)^2(a-b)^2 = ((a+b)(a-b))^2 = (a^2 - b^2)^2\)

The identity simplifies the expression significantly by reducing the complexity of dealing with two squares into the square of a single difference of squares.
When applying this to our exercise where \(a = \sqrt{x}\) and \(b = \sqrt{y}\), the expression simplifies in stages, eventually leading to a polynomial. This process of breaking the product of squares into a sum of terms is a crucial algebraic technique for simplifying and expressing complex polynomial expressions.

Difference of Squares

The "Difference of Squares" formula is a powerful tool in algebra that can quickly transform expressions. This is expressed as:\[(a^2 - b^2) = (a+b)(a-b)\]Where the expression on the right-hand side expands a simple difference of squares into the product of two binomials.
In our original exercise, the intermediate expression is \((\sqrt{x})^2 - (\sqrt{y})^2\). On simplification, it becomes \(x - y\). But note that the expression is in a squared form,

• \((x-y)^2 = ((x-y))((x-y))\)

Understanding the difference of squares helps you recognize and simplify expressions similar to complex polynomial problems and aids in further simplifications by reducing identifiable patterns.

Binomial Expansion

The formula for expanding the square of a binomial is a staple in algebra. The formula is given by:

• \((a-b)^2 = a^2 - 2ab + b^2\)
• \((a+b)^2 = a^2 + 2ab + b^2\)

In the exercise, we expanded \((x-y)^2\) using this formula to yield the result:\[x^2 - 2xy + y^2\]The binomial expansion is particularly useful as it simplifies the polynomial into a manageable form that is easier to interpret and manipulate in subsequent algebraic operations.
Each term in the final expression—\(x^2\), \(-2xy\), and \(y^2\)—comes from specific steps of multiplying terms together, demonstrating the comprehensive power of binomial expansions to open up expressions into sums of simpler terms.

Simplification of Radicals

Simplification of radicals is a fundamental skill required in algebra. It involves reducing complex square roots or other radicals to their simplest form. A radical expression is termed simplified when:

• No perfect square factors other than 1 exist under the radical.
• No fractions remain under the radical sign.
• No radicals appear in the denominator.

In the given problem, \(\sqrt{x}\) and \(\sqrt{y}\) are key radicals. When you square these radicals, \((\sqrt{x})^2\) and \((\sqrt{y})^2\) directly simplify to \(x\) and \(y\), respectively.
This simplification is crucial since it allows us to transform the initial expression into a polynomial. Understanding how to simplify radicals allows you to better handle a broad range of mathematical problems and ensures clarity throughout complex algebraic computations.