Problem 40 - College Algebra

Question

Factor the difference of two squares. $$ x^{2}-144 $$

Step-by-Step Solution

Answer

The factored form of the given expression \(x^{2}-144\) is \((x-12)(x+12)\).

Step 1: Identify the Square Terms

The given polynomial expression is \(x^{2}-144\). In this, the square terms are \(x^{2}\) and \(144\). The square root of \(x^{2}\) is |x| and the square root of \(144\) is 12.

Step 2: Apply the Difference of Squares Formula

The formula for factoring the difference of squares is \(a^{2} - b^{2} = (a-b)(a+b)\). In the given expression, 'a' is 'x' and 'b' is '12'.

Step 3: Substitute the Values

Plug the identified values into the formula to get it factored. We get \((x-12)(x+12)\).

Key Concepts

Difference of Squares

The concept of the difference of squares is a common pattern in algebra that focuses on expressing certain polynomial expressions in a simplified form. This is particularly useful because it allows us to factor expressions and solve equations more easily. The difference of squares takes the form \(a^2 - b^2\), where both \(a\) and \(b\) are perfect squares. Here's what you need to know about this key concept:

• The expression itself is a subtraction, indicated by "difference," of two squares.
• Examples of square numbers include \(4, 9, 16, 25,\) and so on, representing \(2^2, 3^2, 4^2,\) and \(5^2\).
• Recognizing expressions like \(x^2 - 144\) as a difference of squares is crucial. In this case, \(x^2\) and \(144\) (since \(12^2 = 144\)) meet the criteria.

Once you've identified both squares, use the formula \((a - b)(a + b)\), which simplifies the expression by factoring it into two binomials.

Polynomial Expressions

Polynomial expressions are an indispensable part of algebra. They consist of variables and coefficients combined using arithmetic operations: addition, subtraction, multiplication, and non-negative integer exponents of variables. These can be

• One-term, called monomials, like \(3x\).
• Two-term, known as binomials, such as \(x^2 - 9\).
• Multi-term expressions are called polynomials, such as \(x^2 - 144\).

When working with polynomials, understanding how to categorize and manipulate these expressions, such as factoring them, is key. For instance, \(x^2 - 144\) is a polynomial expression because it involves variables and constants where the highest power of the variable is 2, making it a quadratic form.

Quadratic Equations

Quadratic equations are second-degree polynomial equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). Solving quadratic equations can involve various methods, including factoring, completing the square, or using the quadratic formula. When we talk about equations like \(x^2 - 144 = 0\), we see a quadratic equation in its simplest form:

• This specific equation lacks the \(bx\) term, often making it a "difference of squares" scenario.
• Factoring, as shown with \(x^2 - 144\) results in \((x-12)(x+12)\).
• This solves the equation by revealing that \(x = 12\) or \(x = -12\) are solutions, showcasing the importance of recognizing and applying patterns like the difference of squares.

Understanding the link between polynomial expressions and quadratic equations helps unlock efficient solving techniques for complex algebraic problems.