Problem 5 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. \(x^{2}+2 x y+y^{2}=(\quad+\quad)^{2}\) b. \(x^{2}-2 x y+y^{2}=(x \quad y)^{2}\) \(-\quad)\) c. \(x^{2}-y^{2}=(x \quad y)(\quad-\quad)\)

Step-by-Step Solution

Answer

a: \((x+y)^2\); b: \((x-y)^2\); c: \((x+y)(x-y)\).

Step 1: Recognize the Pattern

For part a, notice the expression is a square trinomial, specifically a perfect square trinomial. The standard form for a perfect square is \((x + y)^2\), which expands to \(x^2 + 2xy + y^2\).

Step 2: Solve Part a

To fill in the blanks for part a, recognize that the given expression can be rewritten using the perfect square trinomial identity. Therefore, \(x^{2}+2xy+y^{2}=(x+y)^{2}\).

Step 3: Recognize the Pattern for Part b

Part b also follows the pattern of a perfect square, but this time it's a difference of squares. The expression \((x-y)^2\) expands to \(x^2 - 2xy + y^2\).

Step 4: Solve Part b

To fill in the blanks for part b, recognize that the expression can be rewritten as \(x^2 - 2xy + y^2 = (x - y)^2\).

Step 5: Recognize the Difference of Squares Pattern

Part c is an example of a difference of squares, which follows the formula \(x^2 - y^2 = (x+y)(x-y)\).

Step 6: Solve Part c

To fill in the blanks for part c, use the difference of squares formula to rewrite the expression as \(x^2 - y^2 = (x + y)(x - y)\).

Key Concepts

Difference of Squares

The Difference of Squares is a neat trick in algebra that helps when you have two perfect squares being subtracted from one another. It's an algebraic identity and is useful for simplifying expressions and solving equations. The formula for the difference of squares is \(x^2 - y^2 = (x+y)(x-y)\). This formula indicates that the difference between two squared numbers can be represented as a product of two binomials.

Here's why this works: When you multiply \((x+y)\) by \((x-y)\), the middle terms \(xy - xy\) cancel each other out, leaving you with \(x^2 - y^2\).

• It is applicable when both terms in the expression are perfect squares.
• Speeds up factoring processes in algebraic calculations.

Recognizing this pattern helps in solving and simplifying polynomial expressions, especially when dealing with quadratic equations.

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved. They form the core of many algebraic operations and are essential for simplifying expressions and solving problems.

Some common algebraic identities include:

• The Perfect Square Trinomials: \((x + y)^2 = x^2 + 2xy + y^2\) and \((x - y)^2 = x^2 - 2xy + y^2\).
• The Difference of Squares: \(x^2 - y^2 = (x + y)(x - y)\).

These identities provide shortcuts in calculations and help understand the structure of polynomial expressions. In practice, they simplify complicated expressions and solve equations faster by recognizing patterns and rearranging terms efficiently.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving a sum of powers in one or more variables with coefficients. A simple polynomial could look like \(3x^2 + 2x - 5\). In the context of Perfect Squares or Difference of Squares, understanding polynomial expressions is crucial.

Here's what makes up polynomial expressions:

• **Terms:** Parts of an expression separated by plus or minus signs. Each term contains a coefficient and a variable.
• **Degree:** The highest power of the variable in the expression. A quadratic expression like \(x^2 + 2x + 1\) has a degree of 2.
• **Coefficients:** Numbers multiplying the variables, like the 3 in \(3x^2\).

Polynomial expressions can be simplified and solved using identities like the Perfect Square or Difference of Squares, making algebraic problem-solving more manageable and systematic. Recognizing each component’s role aids in manipulating and solving them efficiently.