Problem 23 - College Algebra

Question

For the following exercises, factor the polynomial. $$ 16 x^{2}-100 $$

Step-by-Step Solution

Answer

The factored form is \( (4x - 10)(4x + 10) \).

Step 1: Identify as a Difference of Squares

The polynomial is in the form \( a^2 - b^2 \), which is a difference of squares. Here, \( a = 4x \) and \( b = 10 \), since \( (4x)^2 = 16x^2 \) and \( 10^2 = 100 \).

Step 2: Apply the Difference of Squares Formula

The formula for the difference of squares is \( a^2 - b^2 = (a - b)(a + b) \). Apply this formula using \( a = 4x \) and \( b = 10 \) to factor the polynomial.

Step 3: Substitute and Simplify

Substitute \( a = 4x \) and \( b = 10 \) into the formula: \( (4x - 10)(4x + 10) \). Thus, the polynomial \( 16x^2 - 100 \) is factored as \( (4x - 10)(4x + 10) \).

Key Concepts

Difference of Squares

The concept of "difference of squares" is a special algebraic pattern used in factoring certain polynomials. It is recognized by its specific form: \( a^2 - b^2 \). This expression represents a difference (subtraction) between two perfect squares. Identifying this pattern is crucial in simplifying and solving algebraic expressions.
When you spot a difference of squares, it can be easily factored using the identity:

• \( a^2 - b^2 = (a - b)(a + b) \)

No matter what numbers or expressions \( a \) and \( b \) represent, this formula will always break the expression into its two factors. Applying this concept not only simplifies equations efficiently but also enhances understanding of these fundamental properties of numbers. Recognizing this pattern allows for quick factoring, saving time and making complex algebraic manipulations easier.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler polynomial factors. This process is fundamental to solving polynomial equations, simplifying expressions, and is widely used in calculus and higher algebra.
The basic idea of factoring is to decompose a complex polynomial into parts that, when multiplied together, give the original polynomial. In the given exercise \( 16x^2 - 100 \), we identify it as a difference of squares, allowing us to apply the formula directly.
To factor polynomials effectively:

• Identify common patterns like difference of squares, perfect square trinomials, or cube formulas.
• Look for common factors in all terms, using them to simplify first if possible.
• Apply appropriate factoring techniques or identities.

The process helps not only in simplifying expressions but also in solving equations when set to zero, as factoring gives the potential solutions or roots of the polynomial equation.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and mathematical operations like addition, subtraction, multiplication, and division. They serve as the language of algebra, representing real-world problems and abstract mathematical ideas.
Understanding and manipulating algebraic expressions is key to solving equations and understanding algebra concepts. In the expression \( 16x^2 - 100 \), both terms are specific types of algebraic components:

• \( 16x^2 \) is a term composed of a coefficient (16) and a variable raised to an exponent (\( x^2 \)).
• 100 is a constant term.

Manipulating algebraic expressions involves understanding their structure and applying algebraic identities like difference of squares, as seen in the exercise. Mastery over algebraic expressions enables developers and engineers to model and solve numerical problems, making it a cornerstone of mathematical literacy.