Problem 4 - Algebra 1: Concepts and Skills

Question

Use a special product pattern to find the product. $$ (w+11)(w-11) $$

Step-by-Step Solution

Answer

The product of the expression \((w+11)(w-11)\) is \(w^2 - 121\).

Step 1: Identify the Pattern

The given expression \((w+11)(w-11)\) can be recognized as the difference of squares pattern, where \(a = w\) and \(b = 11\).

Step 2: Apply the Difference of Squares Formula

We substitute \(a\) and \(b\) into the formula \(a^2 - b^2\). Thus, this simplifies to \(w^2 - 11^2\).

Step 3: Calculate the Result

Solve for \(11^2\) which equals 121, and rewrite the formula accordingly to get \(w^2 - 121\).

Key Concepts

Special Product Patterns

Understanding special product patterns in algebra is crucial as it allows us to simplify complex problems quickly. A common pattern is the difference of squares, which is an expression of the form \(a^2 - b^2\). This is considered a special product because it corresponds to the multiplication of a pair of binomials that are exact opposites—namely, \(a + b\) and \(a - b\).

When you come across products like \(w + 11\) and \(w - 11\), you can instantly recognize them as candidates for the difference of squares pattern. Memorizing this pattern can save a great deal of time in both factoring and expanding algebraic expressions.

Algebraic Expressions

At the core of many mathematical problems are algebraic expressions. An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations. In the context of the difference of squares, we look at expressions comprised of two terms, such as \(a^2 - b^2\).

Variables like 'w' in the exercise \(w + 11\) represent unknown values and are an integral part of algebra. It is crucial to grasp how variables interact with numbers and operations to form expressions that we can manipulate to solve equations.

Polynomial Factoring

In the realm of algebra, polynomial factoring is akin to breaking down a composite number into its prime factors. It involves expressing a polynomial as a product of its factors. Factoring is indispensable for simplifying expressions and solving equations.

Using the special product patterns can make factoring much more manageable. For instance, identifying the expression \(w^2 - 121\) as a difference of squares immediately allows us to factor it into \(w - 11\) and \(w + 11\). It is a simple yet powerful technique that can simplify more complex polynomial expressions.

Simplifying Algebraic Expressions

The process of simplifying algebraic expressions often involves reducing them to a more manageable form. One way to do this is by employing the difference of squares method. By recognizing expressions that fit into this pattern, we can simplify without extensive computation.

For example, in our exercise, we have \(w+11\) and \(w-11\). Instead of multiplying the two binomials using the FOIL method, we simplify by recognizing the pattern and using the corresponding formula \(a^2 - b^2\). This technique results in a final expression that is much easier to work with, both for further algebraic manipulation and for plugging in numerical values.