Problem 58 - Elementary and Intermediate Algebra

Question

Factor. $$ 2 a^{2}-200 b^{2} $$

Step-by-Step Solution

Answer

The expression is factored as \(2(a - 10b)(a + 10b)\).

Step 1: Identify Common Factors

First, look for any common factors in the terms of the expression. The expression is \(2a^2 - 200b^2\). Both terms have a common factor of 2.

Step 2: Factor Out the Common Factor

Factor out the 2 from the expression: \(2(a^2 - 100b^2)\). Now we have a simpler expression inside the parentheses.

Step 3: Recognize the Difference of Squares

The expression \(a^2 - 100b^2\) is a difference of squares. It follows the pattern \(x^2 - y^2 = (x - y)(x + y)\). Identify \(x = a\) and \(y = 10b\) since \(100b^2 = (10b)^2\).

Step 4: Apply the Difference of Squares Formula

Using the difference of squares formula, factor \(a^2 - 100b^2\) as \((a - 10b)(a + 10b)\).

Step 5: Combine All Factors

Combine the factor from Step 2 with the factors from Step 4: \(2(a - 10b)(a + 10b)\). This is the completely factored form of the original expression.

Key Concepts

Difference of Squares

The concept of the 'difference of squares' is an important technique in algebra used to factorize certain expressions. When you see an expression formatted as the difference of two squares, like \( x^2 - y^2 \), it can always be factored into the product of two binomials: \( (x - y)(x + y) \).
This pattern arises because when these two binomials are expanded (multiplied together), the middle terms cancel each other out, leaving the difference of squares.
In our example, the expression \( a^2 - 100b^2 \) fits this formula:

• Here, \( a^2 \) is the first square, represented by \( x^2 \).
• \( 100b^2 \) is the second square, represented by \( y^2 = (10b)^2 \).
• So using the pattern, the expression factors to \( (a - 10b)(a + 10b) \).

Recognizing this pattern makes factoring faster and simplifies many algebra problems.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and subtraction). They are the foundation of algebra. Understanding how to manipulate and factor algebraic expressions is key to solving more complex algebraic equations.
For example, in the expression given: \( 2a^2 - 200b^2 \), we apply algebraic techniques to simplify it.
The process involves identifying patterns and specific properties in the expression. The initial expression is composed of two terms:

• The first term is \( 2a^2 \), and
• the second term is \(-200b^2 \).

Using algebraic manipulation, these expressions can be paired with various factorization techniques like finding common factors and recognizing the difference of squares. Understanding these underlying structures within expressions is essential to mastering algebra.

Common Factors

Finding common factors in a polynomial is often the first step in simplification and factorization. A common factor is a number or expression that divides exactly into each term of the polynomial.
The expression \( 2a^2 - 200b^2 \) initially seems complicated, but by identifying and removing the common factor, we simplify the problem significantly.
In this example:

• Each term (\( 2a^2 \) and \( -200b^2 \)) shares a common factor of 2.

By factoring out this common factor, the expression becomes \( 2(a^2 - 100b^2) \).
This simplification reduces the complexity of the expression and brings it closer to a recognizable form that can be further factored, as seen with the difference of squares method. Identifying and factoring out common factors is a crucial step that reveals the true structure of algebraic expressions, making them more manageable.