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Chapter 1: Problem 10

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

Short Answer

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

Step by step solution

01

Expand the First Expression

Start by expanding \( (\sqrt{x} + \sqrt{y})^2 \) using the formula for a perfect square: \[ (a + b)^2 = a^2 + 2ab + b^2 \] Thus, \( (\sqrt{x} + \sqrt{y})^2 = x + 2\sqrt{x}\sqrt{y} + y \).
02

Expand the Second Expression

Similarly, expand \( (\sqrt{x} - \sqrt{y})^2 \) using the formula for a perfect square: \[ (a - b)^2 = a^2 - 2ab + b^2 \] Thus, \( (\sqrt{x} - \sqrt{y})^2 = x - 2\sqrt{x}\sqrt{y} + y \).
03

Multiply the Expanded Expressions

Now multiply the two expanded results from Steps 1 and 2: \[ (x + 2\sqrt{x}\sqrt{y} + y)(x - 2\sqrt{x}\sqrt{y} + y) \].
04

Apply the Difference of Squares Formula

Recognize the expression as a difference of squares: \[ (a+b)(a-b) = a^2 - b^2 \] Let \( a = x + y \) and \( b = 2\sqrt{x}\sqrt{y} \). Thus, \[ (x + y)^2 - (2\sqrt{x}\sqrt{y})^2 \].
05

Simplify Each Term

Calculate each part of the expression: \( (x+y)^2 = x^2 + 2xy + y^2 \) and \( (2\sqrt{x}\sqrt{y})^2 = 4xy \).
06

Combine the Expressions

Subtract the squared terms: \[ x^2 + 2xy + y^2 - 4xy \]. Simplify it to: \[ x^2 - 2xy + y^2 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expanding Expressions
Expanding expressions is a crucial step in dealing with polynomial expressions. It involves multiplying out expressions enclosed in parentheses to simplify or reformat them. Let's take the expression
  • \( (\sqrt{x} + \sqrt{y})^2 \).
Using the formula for a perfect square, which states
  • \((a + b)^2 = a^2 + 2ab + b^2\),
we expand it to
  • \(x + 2\sqrt{x}\sqrt{y} + y\).
The same process applies to
  • \((\sqrt{x} - \sqrt{y})^2\),
resulting in
  • \(x - 2\sqrt{x}\sqrt{y} + y\).
Expanding allows you to see all the parts of the expression clearly and prepares you for further operations like multiplication or factoring.
Perfect Square Formulas
The perfect square formula is an essential tool for simplifying algebraic expressions. It helps in expanding expressions like \((a + b)^2\) or \((a - b)^2\). These formulas are comprised of three main parts:
  • \(a^2\)
  • \(2ab\)
  • \(b^2\)
For instance, expanding \((\sqrt{x} + \sqrt{y})^2\) using the formula gives us
  • \(x + 2\sqrt{x}\sqrt{y} + y\).
When we deal with \((\sqrt{x} - \sqrt{y})^2\), it is adjusted to
  • \(x - 2\sqrt{x}\sqrt{y} + y\).
Applying these formulas streamlines the process and ensures precision in algebraic manipulations.
Difference of Squares
The difference of squares is a mathematical formula used to factor expressions. It simplifies the multiplication of terms such as
  • \((a+b)(a-b)\)
into the more manageable
  • \(a^2 - b^2\).
In our example, after expanding our perfect squares, we identify it as a difference of squares with
  • \((x + 2\sqrt{x}\sqrt{y} + y)(x - 2\sqrt{x}\sqrt{y} + y)\).
This can be transformed into
  • \((x+y)^2 - (2\sqrt{x}\sqrt{y})^2\).
Understanding how to apply the difference of squares is vital for efficient problem-solving and simplifying complex algebraic expressions.
Simplifying Expressions
The final step in handling polynomial expressions usually involves simplification. After expanding and applying the difference of squares, you are left with a more complex expression that appears as
  • \((x+y)^2 - (2\sqrt{x}\sqrt{y})^2\).
By calculating these components, you find
  • \((x+y)^2 = x^2 + 2xy + y^2\)
  • \((2\sqrt{x}\sqrt{y})^2 = 4xy\).
Subtract the two results to simplify it into
  • \(x^2 - 2xy + y^2\).
This step ensures that the expression is in its simplest possible form, making it easier to interpret and apply in various mathematical contexts.

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Most popular questions from this chapter

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[3]{(2 r-s)^{3}}$$

Simplify the expression. $$\frac{12+r-r^{2}}{r^{3}+3 r^{2}}$$

Simplify the expression. $$\left(x^{2}-4\right)^{1 / 2}(3)(2 x+1)^{2}(2)+(2 x+1)^{3\left(\frac{1}{2}\right)\left(x^{2}-4\right)^{-1 / 2}(2 x)}$$

Simplify the expression. $$\frac{y^{-1}+x^{-1}}{(x y)^{-1}}$$

Rationalize the denominator. $$\frac{1}{\sqrt[3]{x}+\sqrt[3]{y}}$$
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