Problem 6 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ 16-c^{2} $$

Step-by-Step Solution

Answer

The factored form of the expression \(16 - c^{2}\) is \((4 - c)(4 + c)\).

Step 1: Identify 'a' and 'b'

In our expression, \(16 - c^{2}\), \(16\) is a perfect square, and so is \(c^{2}\). Therefore, we can let \(a = \sqrt{16} = 4\) and \(b = c\) where \(a^{2} = 16\) and \(b^{2} = c^{2}\).

Step 2: Apply the Difference of Squares Formula

Now that we have identified 'a' to be 4 and 'b' to be 'c', we can apply the difference of squares formula. The formula is \((a - b)(a + b)\) where 'a' is the square root of the first term and 'b' the square root of the second term. Substituting the values gives \((4 - c)(4 + c)\).

Key Concepts

Factoring

Factoring is a technique used in algebra to simplify expressions or solve equations by breaking them down into products of simpler expressions. When you factor an expression, you essentially reverse the multiplication process to find the expression's "building blocks."
For instance, if you have the expression \(16 - c^2\), you're looking to express it as a product of two binomials. The easiest way to spot if an expression can be factored is to look for common patterns or identities such as the difference of squares.

• Step-by-step: start by identifying terms that can be rewritten or taken out of the expression.
• Look for special patterns like sum or difference of squares, trinomial, or a common factor.
• Apply the appropriate formula or technique to factor completely.

Practicing these techniques will allow you to solve algebraic expressions with ease, leading to simplified expressions or possible solutions to equations.

Perfect Squares

Perfect squares play an essential role in algebra and particularly in factoring expressions. A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it equals \(4^2\).
In algebra, variables can also form perfect squares, such as \(c^2\) which represents \(c \times c\). When you see a perfect square in an expression, it may hint that the expression can be factored using the difference of squares or other methods.

• Recognize perfect squares, both numerical and algebraic, to simplify expressions.
• Use them for identifying opportunities to apply special factoring formulas like the difference of squares.

Understanding perfect squares allows you to spot patterns quickly, making the factoring process faster and more intuitive.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and mathematical operations that are combined to express a value or equation. Dealing with these expressions involves manipulating them to simplify or solve beyond their initial form.
In the example \(16 - c^2\), we are working with a simple algebraic expression that can be rewritten using factoring techniques. These expressions can be simple, like a linear term, or complex, involving multiple variables and powers.

• Understand each term and its operation to manipulate expressions correctly.
• Identify parts of the expression that can be rewritten using known identities or formulas.
• Simplify expressions through operations such as grouping, factoring, or expanding.

Mastering algebraic expressions is like learning a language: it enables you to better understand and manipulate the "sentences" of math.