Problem 13 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$(x+2)^{2}-y^{2}$$

Step-by-Step Solution

Answer

\((x + 2 - y)(x + 2 + y)\)

Step 1: Identify the Difference of Squares

The difference of squares pattern is given by the formula \( a^2 - b^2 = (a-b)(a+b) \). In our expression \( (x+2)^2 - y^2 \), we can identify \( a = (x+2) \) and \( b = y \). This means \( a^2 = (x+2)^2 \) and \( b^2 = y^2 \).

Step 2: Apply the Formula

Use the difference of squares formula \( a^2 - b^2 = (a-b)(a+b) \). Substitute \( a \) and \( b \) into the formula. The expression becomes \( ((x+2) - y)((x+2) + y) \).

Step 3: Simplify the Expression

We simplify \( ((x+2) - y)((x+2) + y) \) by writing it as two factors: \( (x + 2 - y)(x + 2 + y) \).

Key Concepts

Factoring

Factoring is a method used in algebra to simplify expressions or solve equations. When we factor an expression, we essentially break it down into simpler parts, known as factors. These factors, when multiplied together, will give you the original expression. In the context of the difference of squares, factoring is the process of finding two identical binomials that can be multiplied to obtain a polynomial.

The difference of squares is a specific type of factoring. It applies to terms that are squared and subtracted from one another. The pattern is quite recognizable and follows the formula:

• \( a^2 - b^2 = (a - b)(a + b) \)

This is very useful because it allows us to take complex expressions and break them down, making them easier to work with, whether for simplifying purposes or solving equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They form the building blocks for algebra and can be manipulated and solved for various values.

Understanding algebraic expressions involves recognizing the components, such as:

• Constants: Numbers that stand alone without any variables, like 5 or -3.
• Variables: Symbols that represent unknowns, typically letters like \( x \) and \( y \).
• Coefficients: Numbers that precede variables, indicating multiplication, like the 3 in \( 3x \).

In our exercise, \( (x+2)^2 - y^2 \) is an algebraic expression. Identifying components is key to manipulating these expressions, such as when applying formulas like the difference of squares.

Working with algebraic expressions often involves operations such as addition, subtraction, and factoring to make them easier to interpret or solve.

Polynomial Identities

Polynomial identities are equations that hold true for all variable values. These identities are powerful tools in algebra because they provide a shortcut to understanding the structure of polynomials without having to expand or simplify each time.

The difference of squares, \( a^2 - b^2 = (a-b)(a+b) \), is one such identity. It tells us that any polynomial that looks like this can be factored directly into two binomials.
In the exercise given, we apply this identity to transform \( (x+2)^2 - y^2 \) into \( (x+2-y)(x+2+y) \).
Identities like these are crucial. They allow us to figure out the relationships between terms quickly.

• They simplify complex expressions.
• They help solve equations more efficiently.
• They reveal hidden patterns within expressions.

Understanding polynomial identities helps in recognizing how different terms relate, which is a key skill in advanced algebra.