Problem 37 - Algebra 2 and Trigonometry

Question

In \(27-39,\) factor each polynomial completely. $$ (c+2)^{2}-1 $$

Step-by-Step Solution

Answer

The factored form is \((c+1)(c+3)\).

Step 1: Recognize the pattern

The expression \((c+2)^{2}-1\) fits the difference of squares pattern, which is \(a^2 - b^2\). In this case, \(a = (c+2)\) and \(b = 1\).

Step 2: Apply the difference of squares formula

The difference of squares formula is \(a^2 - b^2 = (a-b)(a+b)\). Applying this to our expression gives: \[(c+2-1)(c+2+1).\]

Step 3: Simplify the expression

Simplify each part of the factored expression. First, simplify \((c+2-1)\), which results in \((c+1)\), and then simplify \((c+2+1)\), resulting in \((c+3)\).

Step 4: Write the final factored form

The fully factored form of the expression \((c+2)^{2}-1\) is: \[(c+1)(c+3).\]

Key Concepts

Difference of Squares

The difference of squares is a powerful concept in algebra that helps us rewrite certain expressions in simpler forms. It refers to any expression structured as \(a^2 - b^2\), where both \(a\) and \(b\) are squared terms.
Understanding this pattern allows us to quickly factor complex polynomial expressions into more manageable products of binomials.
Here's a fundamental aspect of the difference of squares:

• The difference indicates subtraction between two squares.
• The formula to factor a difference of squares is \(a^2 - b^2 = (a-b)(a+b)\).

This pattern is especially useful as it straightforwardly breaks down polynomials, which might otherwise seem complex. It allows us to simplify calculations and solve equations more efficiently.

Factoring Techniques

Factoring is a technique used to rewrite polynomials as the product of simpler expressions. It's akin to pulling apart a sandwich and seeing its ingredients, allowing us to understand and manipulate polynomial expressions better.
Different methods are available depending on the look of the polynomial.
When dealing with the difference of squares, the process is straightforward. You recognize the pattern, identify squares, and apply the difference of squares formula. But here are some general tips:

• Look for common factors first. It's always good to simplify the polynomial as much as possible initially.
• Identify patterns like difference of squares, perfect square trinomials, or other recognizable forms.
• Practice makes perfect. Like any skill, the more you work with polynomials, the better you’ll get at spotting these patterns quickly.

Arming yourself with a variety of factoring techniques prepares you to deal with all sorts of polynomial challenges effectively.

Polynomial Expressions

Polynomial expressions are combinations of constants, variables, and exponents. They form the backbone of algebra and can range in complexity from the very simple to the highly intricate.
Understanding the structure of polynomial expressions is crucial for successful factoring.
Here are some features of polynomial expressions:

• They consist of terms. Each term includes a coefficient (numerical part) multiplied by a variable raised to an exponent.
• The degree of a polynomial is determined by the highest exponent present in the expression.
• Polynomial expressions may at times be factorable into the product of simpler expressions, making them easier to work with.

Mastering polynomial expressions involves recognizing these structures and applying operations like addition, multiplication, and factoring, allowing you to simplify, solve, and analyze mathematical situations efficiently.