Question

Factor the expression. Tell which special product factoring pattern you used. $$x^{2}-10,000 w^{2}$$

Short answer

The expression \(x^{2}-10,000 w^{2}\) factors to \((x - 100w)(x + 100w)\). The special product factoring pattern used is 'Difference of Two Squares'

Step-by-step solution

Identify the parts

Rewrite the equation in the form of \(a^2 - b^2\), where \(a = x\) and \(b = 100w\). We square these terms individually: \(a^2 = x^2\) and \(b^2 = (100w)^2 = 10,000w^2\). Hence, the expression can be rewritten as \(a^2 - b^2 = x^2 - 10,000w^2\)

Apply the difference of squares formula

Apply the formula for the difference of squares: \(a^2 - b^2 = (a - b)(a + b)\). Substituting ‘a’ with 'x' and ‘b’ with '100w', the equation becomes: \((x - 100w)(x + 100w)\)

Write the final answer

After factoring the expression, we have: \(x^2 - 10,000w^2 = (x - 100w)(x + 100w)\)

Key concepts

Special Product Factoring Patterns

When we work with algebraic expressions, certain products arise so frequently that they are given special names and formulas. One of the most common is the difference of squares, which can be recognized in the form \( a^2 - b^2 \). The pattern follows a simple formula for factoring: \( a^2 - b^2 = (a - b)(a + b) \).

Understanding the difference of squares is crucial because it simplifies complex expressions into easier, multi-term factors. It's one of the several special product factoring patterns used to break down polynomials that would otherwise be quite challenging to factor.

Algebraic Expressions

In algebra, an algebraic expression is a combination of variables, numbers, and operations. For instance, \( x^2 - 10,000w^2 \) is an algebraic expression involving the variables \( x \) and \( w \).

The beauty of algebra is in how it represents general truths. By learning to manipulate these expressions through operations such as factoring, we can solve a wide variety of problems. It also sets the stage for more advanced mathematics, including calculus and beyond.

Factorization in Algebra

In algebra, factorization is the process of breaking down a complex expression into simpler parts (factors) that, when multiplied together, give the original expression. To factor \( x^2 - 10,000w^2 \), we recognize it resembles the formula for difference of squares.

Why is factorization important?

Factorization simplifies algebraic expressions and equations, making them more manageable. It allows us to find solutions to equations, simplify fractions, and perform operations that are not possible with unfactored expressions. It is the cornerstone of solving quadratic equations and plays a crucial role in higher mathematics.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form \( ax^2 + bx + c = 0 \). The equation \( x^2 - 10,000w^2 = 0 \) is a simple form of a quadratic equation, where the expression \( x^2 - 10,000w^2 \) can be seen as the 'quadratic expression'.

To solve such equations, one often needs to factor the quadratic expression. There are various methods, including using the difference of squares factoring pattern, completing the square, or applying the quadratic formula. Knowing each method and when to apply them is fundamental for anyone studying algebra.