Problem 149 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \((z-2)^{2}-9\)

Step-by-Step Solution

Answer

The expression \((z-2)^2 - 9\) factors to \((z - 5)(z + 1)\).

Step 1: Identify the Expression

The expression given is \((z-2)^2 - 9\). First, recognize this as a difference of squares, where \(a^2 - b^2 = (a-b)(a+b)\). In this case, \(a = (z-2)\) and \(b = 3\).

Step 2: Apply the Difference of Squares Formula

Use the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\) to factor the expression. Substitute \(a = (z-2)\) and \(b = 3\):\[(z-2)^2 - 9 = ((z-2) - 3)((z-2) + 3)\].

Step 3: Simplify the Factors

Simplify the factors obtained from applying the difference of squares formula:1. \((z-2) - 3 = z - 2 - 3 = z - 5\)2. \((z-2) + 3 = z - 2 + 3 = z + 1\)Thus, the expression \((z-2)^2 - 9\) factors to \((z - 5)(z + 1)\).

Key Concepts

Difference of Squares

The difference of squares is a special type of mathematical expression. It is written in the form \(a^2 - b^2\). This expression can be factored into \((a - b)(a + b)\). In the exercise provided, we see the expression \((z-2)^2 - 9\). Here, \(a = (z-2)\) and \(b = 3\). The process of recognizing and factoring expressions using the difference of squares makes algebra less complex.

• First, identify terms that can be squared: look for a subtraction between two squared terms.
• Apply the formula \(a^2 - b^2 = (a-b)(a+b)\).
• Simplify the parentheses to arrive at the final factors.

Mastering the difference of squares will allow you to solve equations faster and more efficiently.

Polynomial Factoring

Polynomial factoring is a crucial skill in algebra. It involves breaking down a polynomial into simpler components, or "factors." These factors, when multiplied together, give back the original polynomial. In this context, the factoring of \( (z-2)^2 - 9 \) involved expressing it as a product of simpler polynomials: \( (z - 5)(z + 1) \).

• Start by identifying a familiar pattern or structure, such as the difference of squares.
• Use appropriate factoring techniques, like grouping, the quadratic formula, or special products.
• Double-check by expanding the factors to ensure they match the original polynomial.

By practicing polynomial factoring, students enhance their problem-solving skills and gain confidence in handling more complex algebraic tasks.

College Algebra

College Algebra serves as a foundation for higher-level mathematics courses. It covers a variety of concepts, including factoring, equations solving, and functions. Understanding how to factor polynomials, like in the exercise, is essential for success in College Algebra.

• Key skills include recognizing patterns in expressions and applying the right factoring formulas.
• It involves solving for unknowns by simplifying and reorganizing equations, beginning with foundational expressions.
• Algebraic skills form the backbone of advanced studies in calculus, statistics, and science-related fields.

Grasping these fundamental concepts in College Algebra supports academic growth and opens doors to diverse and exciting career paths.