Problem 45 - A Graphical Approach to College Algebra

Question

Factor each difference of squares completely. $$(a+b)^{2}-16$$

Step-by-Step Solution

Answer

The expression factors to \((a+b-4)(a+b+4)\).

Step 1: Recognize the Difference of Squares

The expression \((a+b)^2 - 16\) is a difference of squares. It can be rewritten as \((a+b)^2 - 4^2\) since 16 is \(4^2\). The difference of squares formula is \(x^2 - y^2 = (x-y)(x+y)\).

Step 2: Identify \(x\) and \(y\)

Here, \(x = a+b\) and \(y = 4\). Using the formula for the difference of squares, we can identify what the terms \(x\) and \(y\) correspond to in our expression.

Step 3: Apply the Difference of Squares Formula

Using the formula \(x^2 - y^2 = (x-y)(x+y)\), substitute \(x = a+b\) and \(y = 4\):\[(a+b)^2 - 4^2 = ((a+b) - 4)((a+b) + 4)\].

Step 4: Simplify the Factored Expression

The expression \((a+b - 4)(a+b + 4)\) is the fully factored form of the original difference of squares \((a+b)^2 - 16\).

Key Concepts

Factoring Techniques

Factoring techniques are essential tools in simplifying polynomial expressions. These techniques allow us to express a polynomial as the product of other polynomials or expressions. They make complex equations much more manageable. For difference of squares specifically, we use a specific formula. This formula is ideal when an expression is composed of two perfect squares with a subtraction sign between them. Here, we are looking at expressions like \(a^2 - b^2\).
When factoring such expressions, we use the difference of squares identity: \(x^2 - y^2 = (x-y)(x+y)\). This formula is derived from the pattern observed in perfect squares.
The steps in factoring include:

• Identifying the terms in the form \(x^2 - y^2\)
• Applying the difference of squares formula
• Rewriting the expression as a product of two binomials

By following these steps, any polynomial expression in the form of a difference of squares can be efficiently factored.

Polynomial Expressions

Polynomial expressions like \((a+b)^2 - 16\) consist of variables raised to whole number powers, and they may include constants as well. Understanding how to manipulate and factor these expressions is crucial in algebra.
In this exercise, the polynomial \((a+b)^2 - 16\) is presented in a structured form that is key for factoring. The expression inside the parentheses, \((a+b)\), is a binomial, and the entire expression is a difference of squares.
To successfully factor polynomial expressions, recognizing patterns like differences of squares is vital. You need to look for:

• Perfect squares in terms or constants
• Binary operations (such as subtraction in this context)

This knowledge helps in selecting the correct factoring technique and whether formulas, like that of the difference of squares, apply.

Algebraic Identities

Algebraic identities are mathematical expressions that are universally true. They are essential shortcuts in algebra that simplify and solve complex polynomial problems.
One such identity is the difference of squares: \(x^2 - y^2 = (x-y)(x+y)\). In this context, understanding and using algebraic identities allows us to transition from recognizing an algebraic form to effectively applying a formula.
For the given problem, \((a+b)^2 - 16\) can be viewed through the lens of this identity. Recognizing the pattern is the first crucial step. We identify the perfect squares \((a+b)^2\) and \(4^2\) and use the exterior parts \((x-y)(x+y)\) to simplify the expression.
Algebraic identities work like a universal key in solving different kinds of polynomials. They not only provide a streamlined method of manipulation but also enhance understanding of how different forms of polynomials relate to one another.