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Chapter 1: Problem 9

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

Short Answer

Expert verified
The expression simplifies to \(x - y\).

Step by step solution

01

Identify the Pattern

Notice that the expression \((\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y})\) fits the difference of squares pattern: \((a + b)(a - b) = a^2 - b^2\). Here, \(a\) is \(\sqrt{x}\) and \(b\) is \(\sqrt{y}\).
02

Apply the Difference of Squares Formula

Using the formula for the difference of squares, substitute \(\sqrt{x}\) for \(a\) and \(\sqrt{y}\) for \(b\) to obtain: \( (\sqrt{x})^2 - (\sqrt{y})^2 \).
03

Simplify the Expression

Calculate \((\sqrt{x})^2\) and \((\sqrt{y})^2\). Recall that the square of a square root cancels out the square root, so \((\sqrt{x})^2 = x\) and \((\sqrt{y})^2 = y\). This simplifies to \(x - y\).

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Key Concepts

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

Polynomials
A polynomial is an algebraic expression consisting of variables and coefficients, connected by addition, subtraction, multiplication, and non-negative integer exponents. They are fundamental building blocks in algebra.
Polynomials can be simple or complex:
  • Simple polynomials: A single variable expression such as \(x^2 + 3x + 2\)
  • Complex polynomials: Multiple variables and higher-degree terms, like \(2x^3y - 3xy + 5\)
Polynomials are used in various mathematical contexts, including solving equations and modeling real-life situations.
Understanding their basic structure and operations is key to excelling in algebra. Recognizing patterns, such as the difference of squares, can simplify polynomial expression exercises significantly.
Simplifying Algebraic Expressions
The goal of simplifying algebraic expressions is to make them easier to understand or solve, often through systematically combining like terms or using algebraic identities.
In the given problem, recognizing the difference of squares pattern helps in simplifying. The expression \((\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y})\) becomes easier when recognizing that it can be expressed as \(a^2 - b^2\).
To simplify:
  • Identify patterns or formulas like the difference of squares: \((a + b)(a - b) = a^2 - b^2\)
  • Substitute the correct terms: In the example, \(a = \sqrt{x}\) and \(b = \sqrt{y}\)
  • Simplify each term: \((\sqrt{x})^2 = x\) and \((\sqrt{y})^2 = y\)
This process reduces the expression to a simpler form, \(x - y\).
The art of simplification helps make complex problems more manageable.
Square Roots
A square root involves finding a number which, when multiplied by itself, results in the given number. The square root of \(x\) is denoted as \(\sqrt{x}\).
Important properties of square roots include that they, by definition, are always non-negative. This is because the product of two negative numbers is positive:
  • \((\sqrt{x})^2 = x\), meaning squaring a square root returns the original number.
  • Square roots are used to find roots of quadratic equations and simplify algebraic expressions.
The given exercise shows this property, where squaring \(\sqrt{x}\) and \(\sqrt{y}\) simplifies the expression to remove the complexity introduced by the root signs.
Understanding square roots and their properties is crucial for algebra and higher-level math, helping solve equations involving roots and simplifying expressions.

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Most popular questions from this chapter

Simplify the expression. $$(6 x-5)^{3}(2)\left(x^{2}+4\right)(2 x)+\left(x^{2}+4\right)^{2}(3)(6 x-5)^{2}(6)$$

Simplify the expression. $$\frac{2}{x}+\frac{3 x+1}{x^{2}}-\frac{x-2}{x^{3}}$$

Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning Give a reason for your answer. $$a^{x} b^{y} \square(a b)^{x y}$$

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

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt{5 x y^{2}} \sqrt{15 x^{3} y^{3}}$$
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