Problem 10 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recognize Identities

The expression given is \((\sqrt{x} + \sqrt{y})^2 (\sqrt{x} - \sqrt{y})^2\). Notice that each factor is a perfect square, and we can use the identity \((a+b)^2 = a^2 + 2ab + b^2\) and \((a-b)^2 = a^2 - 2ab + b^2\).

Step 2: Expand Each Factor

First, expand \((\sqrt{x} + \sqrt{y})^2\):\[(\sqrt{x})^2 + 2\sqrt{x}\sqrt{y} + (\sqrt{y})^2 = x + 2\sqrt{xy} + y\]Now, expand \((\sqrt{x} - \sqrt{y})^2\):\[(\sqrt{x})^2 - 2\sqrt{x}\sqrt{y} + (\sqrt{y})^2 = x - 2\sqrt{xy} + y\]

Step 3: Multiply the Expanded Expressions

Multiply the results of the previous step:\[(x + 2\sqrt{xy} + y)(x - 2\sqrt{xy} + y)\]Use the distributive property here, treating it as a product of binomials: \((a+b)(a-b) = a^2 - b^2\).Recognizing this is a difference of squares:\[a = x+y, \quad b = 2\sqrt{xy}\]Substitute these into the difference of squares:\[(x+y)^2 - (2\sqrt{xy})^2\]

Step 4: Simplify the Expression

First calculate the squares:\[(x+y)^2 = x^2 + 2xy + y^2\]\[(2\sqrt{xy})^2 = 4xy\]Substitute these results back:\[x^2 + 2xy + y^2 - 4xy\]Simplify the expression:\[x^2 + 2xy - 4xy + y^2 \Rightarrow x^2 - 2xy + y^2\]

Step 5: Conclusion

The polynomial expression is found to be:\[x^2 - 2xy + y^2\] This is the simplified polynomial form of the given expression.

Key Concepts

Difference of Squares

In algebra, the difference of squares is a very useful concept used to simplify expressions. It is an algebraic identity that states that any two terms a and b satisfy:

• \((a+b)(a-b) = a^2 - b^2\)

In the problem provided, when multiplying the two binomials, we recognize that they form a difference of squares. This allows us to simplify the binomial product \((x+2\sqrt{xy} + y)(x-2\sqrt{xy} + y)\). By treating \(x+y\) as one entity and \(2\sqrt{xy}\) as another, we simplify the math to \((x+y)^2 - (2\sqrt{xy})^2\). This step reduces complexity and paves the way for easier handling of the expression.

Expanding Binomials

Expanding binomials is essential when working with algebraic expressions. Applying the binomial square identities allows us to expand them fully.

• The identity for expanding \((a+b)^2\) is given by: \(a^2 + 2ab + b^2\)
• Similarly, for \((a-b)^2\), the identity is: \(a^2 - 2ab + b^2\)

In the exercise, we first expand \((\sqrt{x} + \sqrt{y})^2\) and \((\sqrt{x} - \sqrt{y})^2\) using these identities:

• For \((\sqrt{x} + \sqrt{y})^2\): \(x + 2\sqrt{xy} + y\)
• For \((\sqrt{x} - \sqrt{y})^2\): \(x - 2\sqrt{xy} + y\)

Each expansion opens the way to a more comprehensive solution and leads into the multiplication of these expanded expressions.

Algebraic Identities

Algebraic identities are useful tools for simplifying expressions, serving as shortcuts for more complicated calculations. Knowing when and how to apply them can save heaps of time.

The identities applied in this problem include:

• The formula for the square of a sum: \((a+b)^2 = a^2 + 2ab + b^2\)
• The formula for the square of a difference: \((a-b)^2 = a^2 - 2ab + b^2\)

Utilizing these allows efficient expansion of polynomial expressions. Recognizing that the multiplication product \((\sqrt{x}+\sqrt{y})^2(\sqrt{x}-\sqrt{y})^2\) incorporates these identities simplifies the process.

In essence, these identities unlock a straightforward means to expand and simplify various algebraic expressions by breaking them into less complex components.

Simplification of Expressions

Simplifying algebraic expressions is akin to cleaning up math equations to make them less cluttered and easier to handle. The end goal is to express the solution in the simplest form possible.

In this exercise, having expanded the polynomial expressions, we simplify the result by combining like terms and subtracting similar values. After calculating \((x+y)^2\) and \((2\sqrt{xy})^2\), we substitute:

• \((x+y)^2 = x^2 + 2xy + y^2\)
• \((2\sqrt{xy})^2 = 4xy\)

Upon placing these calculations back into our initial equation: \(x^2 + 2xy + y^2 - 4xy\), it further resolves to \(x^2 - 2xy + y^2\).

Simplification aids in making sure expressions are as clear and as concise as they can be, bringing order to complex algebraic manipulations.