Problem 1 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$x^{2}-1$$

Step-by-Step Solution

Answer

The factorization of \(x^2 - 1\) is \((x - 1)(x + 1)\).

Step 1: Identify the Difference of Squares Pattern

The pattern for difference of squares is given by \( a^2 - b^2 = (a - b)(a + b) \). In the expression \( x^2 - 1 \), notice that both \( x^2 \) and \( 1 \) are perfect squares. Recognize this pattern to use it for factoring.

Step 2: Express Each Term as a Square

Rewrite \( x^2 \) as \( (x)^2 \) and \( 1 \) as \( (1)^2 \). This can be clearly seen as \( (x)^2 - (1)^2 \), fitting into the difference of squares pattern \((a^2 - b^2)\).

Step 3: Apply the Difference of Squares Formula

Using the formula \( a^2 - b^2 = (a - b)(a + b) \), substitute \( a = x \) and \( b = 1 \) in the expression \( (x)^2 - (1)^2 \). This results in \((x - 1)(x + 1)\).

Step 4: Verification of Factored Expression

Expand \((x - 1)(x + 1)\) to verify that it matches the original expression \( x^2 - 1 \).\((x - 1)(x + 1) = x^2 + x - x - 1 = x^2 - 1.\)The expression correctly expands back to the original polynomial, confirming the factorization is correct.

Key Concepts

Factoring

Factoring is a fundamental aspect of algebra that involves breaking down algebraic expressions into simpler components or "factors." These factors, when multiplied together, give back the original expression. Think of it as reverse multiplication! In the context of polynomial expressions like \( x^2 - 1 \), factoring helps simplify the expression for further manipulation or solving equations.

One common technique in factoring is the "difference of squares" method, used here with \( x^2 - 1 \). This particular pattern, \( a^2 - b^2 = (a-b)(a+b) \), is essential because it turns a difference of perfect squares into a product of two binomials. This makes equations easier to solve or evaluate.

To effectively factor using this pattern:

• Identify any terms in the expression that are perfect squares.
• Rewrite those terms as squares of simpler expressions.
• Utilize the difference of squares formula.

Remember to always check your work by expanding your factored expression to ensure it matches the original!

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations like addition, subtraction, multiplication, and division. These form the building blocks of algebra and allow us to represent mathematical relationships in a symbolic format.

In the expression \( x^2 - 1 \), we see a simple algebraic expression that involves both a variable \( x \) and a numerical constant. The operations here are especially important as they dictate which factoring technique, like the difference of squares, can be employed.

When working with algebraic expressions:

• Always look for common patterns, such as perfect squares or like terms.
• Consider rewriting expressions in an expanded form to identify these patterns.

This approach not only simplifies solving equations but also helps in understanding the structure and properties of the expression involved.

Polynomials

Polynomials are algebraic expressions consisting of variables raised to whole number powers, coefficients (which are numbers), and involve the operation of addition or subtraction. They can have one or multiple terms.

The expression \( x^2 - 1 \) is an example of a polynomial, specifically a quadratic polynomial because it features a variable raised to the second power. Quadratics often have various factoring options, and recognizing these can greatly simplify solving or managing the polynomial.

A few key points about polynomials:

• The degree of a polynomial is the largest exponent of its variable(s). For \( x^2 - 1 \), the degree is 2.
• Polynomials can be identified by counting the number of terms: this example, \( x^2 - 1 \), is a binomial as it has two terms.

Understanding the nature of polynomials and their structure is crucial when applying factoring techniques or solving polynomial equations. This foundational knowledge empowers students to tackle more challenging algebraic expressions with confidence.