Problem 45 - College Algebra

Question

For the following exercises, multiply the polynomials. $$ (a+b)(a-b) $$

Step-by-Step Solution

Answer

The result is \(a^2 - b^2\).

Step 1: Recognize the formula type

This expression \((a+b)(a-b)\) is a difference of squares formula. Recall that the difference of squares formula states \((x+y)(x-y) = x^2 - y^2\).

Step 2: Identify the terms

In the given expression, assign \(a\) as \(x\) and \(b\) as \(y\). Thus, \(a+b\) is \(x+y\) and \(a-b\) is \(x-y\).

Step 3: Apply the difference of squares formula

Using the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\).

Step 4: Write the result

The product of \((a+b)(a-b)\) when applying the difference of squares is \(a^2 - b^2\).

Key Concepts

Difference of Squares

Let's explore how the difference of squares works. Imagine it as a shortcut formula in algebra that lets you quickly multiply two specific types of binomials. The concept is based on the expression

• \((x+y)(x-y) = x^2 - y^2\)

This formula is powerful because it simplifies the multiplication process between two terms. Here,

• \((a+b)\) is like \((x+y)\)
• \((a-b)\) is like \((x-y)\)

When you multiply these pairs of binomials, all middle terms cancel each other out. This cancellation is what gives us the result, which is always the first term squared minus the second term squared. Remember, understanding how terms cancel helps in simplifying many algebraic expressions!
Whenever you see expressions like \((a+b)(a-b)\), recognize it as a difference of squares to save time and effort.

Algebraic Expressions

An algebraic expression is a combination of variables and constants joined by mathematical operations such as addition, subtraction, multiplication, or division. In our example of polynomial multiplication, we have

• \((a+b)\) - This expression involves the addition of two terms, \(a\) and \(b\)
• \((a-b)\) - Here, there's a subtraction between \(a\) and \(b\)

These expressions, when multiplied, form a product. Algebraic expressions are foundational in algebra because they represent numbers using symbols.
Understanding the structure of these expressions helps you decide on appropriate formulas to simplify them, such as recognizing when to use the difference of squares or other mathematical formulas. Algebra requires you to manipulate these expressions to solve for specific unknowns or to simplify complex problems.

Mathematical Formulas

Mathematical formulas serve as reliable tools that help us solve problems efficiently. They express relationships or rules between different variables and are essential in simplifying more complex tasks. For instance,

• The difference of squares formula: \((x+y)(x-y) = x^2 - y^2\)

This particular formula streamlines the multiplication of binomials by eliminating the need for detailed distribution. Equations like these are more than just shortcuts; they frame a clear method to work with numbers and expressions.
Remembering key formulas saves you time and enables you to approach algebraic problems with confidence. While learning these formulas, practice identifying when they are applicable. This practice ensures that you're utilizing them efficiently whenever possible, enhancing both speed and accuracy in problem-solving scenarios.