Problem 19 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ a^{2}-100 $$

Step-by-Step Solution

Answer

Answer: The factored form of the expression \(a^2 - 100\) is \((a + 10)(a - 10)\).

Step 1: Identify the Difference of Squares

In the given expression \(a^2 - 100\), we can identify that this is a difference of squares. In this case, \(A = a\) and \(B = 10\) because \(100 = 10^2\). So we have \(A^2 - B^2 = a^2 - 10^2\).

Step 2: Apply the Difference of Squares Formula

Now, we apply the difference of squares formula: \(A^2 - B^2 = (A + B)(A - B)\). In our case, we have \(a^2 - 10^2 = (a + 10)(a - 10)\).

Step 3: Write the Final Answer

The factored form of the given binomial expression is \((a + 10)(a - 10)\).

Key Concepts

Difference of Squares

In algebra, the 'difference of squares' is a specific pattern used to simplify expressions. This technique involves recognizing and factoring expressions of the form \(A^2 - B^2\). The special thing about these expressions is that they can be rewritten in factorized form using the formula: \(A^2 - B^2 = (A + B)(A - B)\).
To break it down:

• A represents one term (such as \(a\) in the exercise), and
• B represents another term (such as \(10\) in the exercise because \(10^2 = 100\)).

In practical terms, if you can identify that both parts of your expression are perfect squares, and they are being subtracted, then you can convert it to the product of a sum and a difference.
This method speeds up calculations and simplifies the problem-solving process because it provides an easy shortcut to find factors quickly.

Binomials

A binomial is a type of algebraic expression that contains exactly two terms. In the expression given in the exercise, \(a^2 - 100\) is considered a binomial. Each term in the binomial is an individual algebraic entity, often separated by a plus (+) or minus (-) sign.
Key things to remember about binomials:

• The two terms could be simple numbers, variables, or more complex expressions like \(a^2\) or \(10^2\).
• Within binomials, operations like addition and subtraction are used to connect the terms.
• They can be factored using techniques like the difference of squares, which reveals hidden relationships between the terms.

Understanding binomials is essential as they often appear in algebraic problems, and recognizing their structure can aid in quickly factorizing or simplifying them.

Algebraic Expressions

Algebraic expressions are foundational to understanding and solving mathematical problems. They are combinations of numbers, variables, and operations like addition, subtraction, multiplication, or division.
The expression \(a^2 - 100\) is a simple example of an algebraic expression. Here:

• \(a^2\) represents a variable raised to a power, showing a squared term.
• \(100\) is a constant, representing a fixed numeric value.
• The subtraction (-) tells us one term is being taken away from the other.

To work with algebraic expressions effectively, it is crucial to understand how to manipulate them through operations like factoring, expanding, and simplifying.
Factoring, like the difference of squares shown in the exercise, helps break down complex expressions into simpler, more manageable parts. Knowing these basics opens up pathways to more advanced algebra topics.