Problem 3 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$16 x^{2}-25$$

Step-by-Step Solution

Answer

The factorization is \((4x - 5)(4x + 5)\).

Step 1: Recognize the Difference of Squares Pattern

The expression given is of the form \( a^2 - b^2 \), which is a difference of squares. This can be factored using the formula \( a^2 - b^2 = (a - b)(a + b) \).

Step 2: Identify 'a' and 'b' Values

We need to express both terms as squares: \( 16x^2 = (4x)^2 \) and \( 25 = 5^2 \). Thus, \( a = 4x \) and \( b = 5 \).

Step 3: Apply the Difference of Squares Formula

Using the values of \( a \) and \( b \) from the previous step, plug them into the difference of squares formula. So, the expression becomes \((4x)^2 - 5^2 = (4x - 5)(4x + 5)\).

Step 4: Verify the Factored Form

Expand \((4x - 5)(4x + 5)\) to ensure it equals the original expression. Applying the distributive property: \( (4x - 5)(4x + 5) = 4x \cdot 4x + 4x \cdot 5 - 5 \cdot 4x - 5 \cdot 5 = 16x^2 - 25 \). This confirms the factorization is correct.

Key Concepts

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler "factor" polynomials that, when multiplied together, give the original polynomial. This is much like breaking down a whole number into its prime factors. There are various methods to factor polynomials, but one of the most common and efficient methods is recognizing special patterns. The **difference of squares** is a specific type of polynomial that can be factored using a straightforward pattern. More often than not, an expression like the one in the exercise, such as \(16x^2 - 25\), fits the difference of squares pattern. Once identified, you can quickly apply the formula: if \( a^2 - b^2 = (a - b)(a + b) \), then both "a" and "b" must be determined from the given expression. This saves both time and effort in simplifying polynomials.

Polynomial Identities

Polynomial identities are equations that hold true for any value substituted into the variables of a polynomial. One of the most applicable identities in algebra is the **difference of squares identity**. This identity is invaluable because it allows for the simplification of polynomial expressions quickly and reliably. In our example, the expression \(16x^2 - 25\) fits this identity perfectly, as it can be rewritten in the form \( (4x)^2 - 5^2 \). Recognizing this allows us to use the identity \( a^2 - b^2 = (a-b)(a+b) \) to factor the polynomial. Applying this identity simplifies algebraic expressions and helps in solving equations. Identifying these patterns enhances your problem-solving toolkit in algebra.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and their operations (such as addition, subtraction, multiplication, and division). Mastering the manipulation of these expressions is foundational in algebra. Manipulating expressions with complex operations becomes more manageable when you understand how to factor and simplify them using identities like the difference of squares. For example, the expression \(16x^2 - 25\) involves recognizing the terms as perfect squares and employing the difference of squares formula. When dealing with algebraic expressions, practicing the identification and application of such patterns will develop a stronger grasp of algebraic concepts and improve your capability to tackle more complex problems efficiently.