Problem 15 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$4 x^{2}-(y+1)^{2}$$

Step-by-Step Solution

Answer

\((2x + y + 1)(2x - y - 1)\)

Step 1: Recognize the Difference of Squares

The expression given is \(4x^2 - (y+1)^2\). Notice that it is in the form \(a^2 - b^2\), where \(a^2 = (2x)^2\) and \(b^2 = (y+1)^2\). The formula for the difference of squares is \(a^2 - b^2 = (a+b)(a-b)\).

Step 2: Identify \(a\) and \(b\)

From the expression \(4x^2 - (y+1)^2\), we can identify \(a = 2x\) and \(b = (y+1)\). These are the terms whose squares form the original expression.

Step 3: Apply the Difference of Squares Formula

Substitute \(a = 2x\) and \(b = (y+1)\) into the formula \(a^2 - b^2 = (a+b)(a-b)\). This gives us:\[(2x + (y+1))(2x - (y+1))\].

Step 4: Simplify the Factored Form

Simplify the expression inside the parentheses:- For the first term: \(2x + (y + 1) = 2x + y + 1\).- For the second term: \(2x - (y + 1) = 2x - y - 1\).Thus, the factored form is \((2x + y + 1)(2x - y - 1)\).

Key Concepts

Polynomial Factoring

Polynomial factoring is like breaking down a number into its prime components, but for polynomials. When you're dealing with polynomial expressions, factoring them can simplify problems and make them easier to solve. In this exercise, we focus on a specific type of polynomial factoring known as the "difference of squares."

• Polynomials are expressions that contain variables raised to varying powers and have coefficients.
• Factoring polynomials involves writing them as a product of simpler expressions or polynomials.
• It’s an essential skill because it helps in simplifying expressions and solving equations more efficiently.

When you see a term like \(4x^2 - (y+1)^2\), you're looking at an expression that can be factored using special patterns like the difference of squares, which makes it simpler to work with in algebraic operations like solving equations or integrating in calculus.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are fundamental to algebra and appear in a variety of contexts, from simple equations to complex polynomials.

• Understanding algebraic expressions means recognizing how to manipulate them through the rules of algebra.
• In our example, the expression \(4x^2 - (y+1)^2\) includes two terms, each of which is a squared expression.

Variables like \(x\) and \(y\) can stand for numbers and are used in expressions to form algebraic equations. Recognizing how these elements interact, especially when combined through operations like addition or subtraction, is crucial for factoring.

Factoring Techniques

Factoring techniques are methods used to break down complex expressions or numbers into their components. One particularly useful method in algebra is factoring by recognizing the difference of squares.

• The difference of squares is a pattern given by \(a^2 - b^2\), which can be factored into \((a+b)(a-b)\).
• In our example, it identifies that we have \((2x)^2 - (y+1)^2\), allowing us to apply the formula directly.
• Once the expression is factored, it simplifies into \((2x + y + 1)(2x - y - 1)\).

Mastering these techniques involves recognizing patterns within the expression. Techniques like these simplify complex expressions, making problems easier to solve. In practice, factoring effectively can turn a daunting-looking polynomial into something more manageable.