Problem 37 - Elementary and Intermediate Algebra

Question

Factor expression. Factor out any GCF first. \(2 x^{2}-288\)

Step-by-Step Solution

Answer

The expression factors to \(2(x - 12)(x + 12)\).

Step 1: Identify the Greatest Common Factor (GCF)

The expression given is \(2x^2 - 288\). Start by identifying the greatest common factor of these two terms. The GCF will be the highest number that divides both 2 and 288. Here, the GCF of 2 and 288 is 2.

Step 2: Factor out the GCF

Now factor the GCF, which is 2, out from the expression \(2x^2 - 288\). This gives you: \[ 2(x^2 - 144) \]

Step 3: Identify the Difference of Squares

The expression inside the parentheses, \(x^2 - 144\), is a difference of squares. Recall that the difference of squares can be written as \(a^2 - b^2 = (a - b)(a + b)\). Here, \(x^2\) is \(a^2\) and \(144\) is \(b^2\) with \(b = 12\).

Step 4: Factor the Difference of Squares

Applying the difference of squares identity to \(x^2 - 144\), it can be factored as \((x - 12)(x + 12)\). So the expression becomes: \[ 2(x - 12)(x + 12) \]

Key Concepts

Greatest Common Factor

When factoring polynomials, identifying the Greatest Common Factor (GCF) is often the first and most crucial step. The GCF of a group of numbers or terms is the largest number or expression that divides all of them without leaving a remainder. Breaking this down:

• The GCF can be a constant, like in our example where it was 2 for the numbers 2 and 288.
• It ensures we simplify expressions properly by ensuring no larger common factors are overlooked.
• Extracting the GCF as the first step not only simplifies the polynomial but also sets up the remaining expression for easier factoring.

To find the GCF, list the factors of each number and identify the largest factor that appears in both lists. Once identified, factor it out from the entire expression. This simplifies the original polynomial, making future steps more straightforward.

Difference of Squares

The difference of squares is a specific type of polynomial that can be factored into two binomials. This occurs when a polynomial is in the form of \(a^2 - b^2\), where both \(a^2\) and \(b^2\) are perfect squares. It’s a handy pattern because it simplifies expressions quite elegantly.

• In the formula \(a^2 - b^2 = (a - b)(a + b)\), you're essentially splitting the square terms into two binomials.
• Identifying this pattern requires recognizing perfect square terms in the expression, like \(x^2\) and 144 in our solution.
• For instance, when faced with \(x^2 - 144\), recognize \(x^2\) as \((x)^2\) and 144 as \(12^2\).

By applying this knowledge to factor \(x^2 - 144\), you rewrite it using the visualized pattern: \((x - 12)(x + 12)\). It's a fast yet solid technique for breaking down quadratic differences.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. Understanding these is fundamental in algebra as they form the basis of equations and functions.

• An algebraic expression might be as simple as \(2x\) or as complex as \(2(x^2 - 144)\).
• Each part of an expression, such as terms, coefficients, and variables, serves a specific role in calculations and transformations.
• Terms are individual components separated by addition or subtraction, coefficients are numbers in front of variables, and variables represent unknown or varying values.

When working with these expressions, factoring converts them to simpler, more manageable forms, especially by using techniques like finding common factors and recognizing patterns such as the difference of squares. Mastering these skills opens the door to solving equations and understanding advanced mathematical concepts.