Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The polynomial is \(x - y\).

Step 1: Identify the expression

The expression given is \((\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})\). This expression is in the form \((a+b)(a-b)\), which is the difference of squares formula.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula states that \((a+b)(a-b) = a^2 - b^2\). Using this formula, set \(a = \sqrt{x}\) and \(b = \sqrt{y}\). This gives us: \((\sqrt{x})^2 - (\sqrt{y})^2\).

Step 3: Simplify the Squares

Simplify \((\sqrt{x})^2\) to \(x\) and \((\sqrt{y})^2\) to \(y\). So, the expression becomes \(x - y\).

Key Concepts

Difference of Squares

The difference of squares formula is an essential concept in algebra that allows us to simplify and solve certain types of polynomial expressions quickly. It's one of the most useful algebraic identities because it helps break down expressions into simpler, more manageable forms. When you encounter an expression like

• \((a+b)(a-b)\)

you can apply the difference of squares formula:

• \(a^2 - b^2\)

Here, two terms are squared and subtracted from each other, and it's called a 'difference of squares' because it involves subtracting one square term from another.

Consider the algebra equation we were discussing:

• \((\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y})\)

This is a classic example of the difference of squares, where our

• \(a = \sqrt{x}\)
• \(b = \sqrt{y}\)

Therefore, according to the formula, the expression simply becomes \(x - y\). Understanding this concept simplifies many algebraic processes, making problem-solving quicker and more intuitive.

Simplifying Radicals

Radicals can often make equations look more complex than they really are. However, simplifying radicals is crucial to bring out the true elegance of algebraic expressions. A radical is an expression that includes a square root, cube root, or any higher root of a number. In our example,

• \(\sqrt{x}\)
• \(\sqrt{y}\)

are both square roots.

To simplify a radical, focus on eliminating the square root by squaring the expression. For instance:

• \((\sqrt{x})^2\)
• \((\sqrt{y})^2\)

In both cases, the radicals are simplified to

• \(x\)
• \(y\)

respectively.

Sometimes the square roots won't simplify neatly into whole numbers, but often, in the difference of squares, as demonstrated here, they do.

Simplifying radicals is a helpful skill, as it removes complexity and aids in further solving and understanding algebraic expressions.

Algebraic Expressions

Algebraic expressions form the foundation of algebra and include numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, or division). An expression itself doesn't have an equality sign, which differentiates it from an equation. In our case,

• \((\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y})\)

is an expression, not an equation.

Understanding algebraic expressions is crucial for solving mathematical problems. They serve as a language that describes mathematical concepts using symbols and numbers. Each part of an expression plays a role.

For example:

• \(\sqrt{x}\) and \(\sqrt{y}\) are variables representing unknown values within the expression.
• Brackets \(()\) show us how to group parts of the expression for simplification.

Becoming comfortable with algebraic expressions allows you to manipulate and solve them expertly, making advanced mathematics more approachable and less intimidating.