Question

Challenge Problems.$$25 x^{4}-16 y^{6}$$

Short answer

\( (5x^2)^2 - (4y^3)^2 \) factors to \( (5x^2 + 4y^3)(5x^2 - 4y^3) \) using the difference of squares formula.

Step-by-step solution

Identify the type of expression

Recognize the given expression as a difference of two squares. The expression can be written as \( (5x^2)^2 - (4y^3)^2 \).

Apply the difference of two squares formula

Use the formula \( a^2 - b^2 = (a+b)(a-b) \) to factor the expression. Let \( a = 5x^2 \) and \( b = 4y^3 \).

Write the factored form

Use the values of \( a \) and \( b \) to write the binomial factors \( (5x^2 + 4y^3)(5x^2 - 4y^3) \) as the factored form of the original expression.

Key concepts

Algebra

Algebra is like a universal language of mathematics, allowing us to represent numbers and operations with symbols and letters. At its core, algebra deals with solving for unknowns and understanding the relationships between variables. In practice, algebraic expressions can take many forms, such as binomials, trinomials, and higher-degree polynomials. Mastering algebra requires understanding how to manipulate these expressions using various techniques, one of which is polynomial factoring. This process is critical because it simplifies expressions and solves equations by breaking them into more manageable parts.

In the provided exercise, algebra is used to powerfully illustrate the concept of factoring, specifically in regard to the difference of two squares. The initial expression, \(25x^{4}-16y^{6}\), might seem daunting at first glance, but algebra gives us the tools we need to deconstruct it and find its simpler components by recognizing patterns and applying specific rules.

Polynomial Factoring

Polynomial factoring is essentially the reverse of multiplying polynomials. The goal is to take a complex expression and break it down into products of simpler expressions. Polynomial factoring is invaluable when solving equations, simplifying algebraic expressions, and analyzing functions. One common pattern students learn to recognize is the \(a^2 - b^2\) pattern, known as the difference of two squares.

Our example, \(25x^{4}-16y^{6}\), is disguised as a complex polynomial, but it's really a difference of two squares in a more elaborate form. Polynomials with four terms or more might be factored by grouping, but in cases like the example where each term is a perfect square separated by a minus sign, the difference of two squares formula is the most efficient method. By following this approach, complicated expressions become products of simpler, more approachable binomial expressions.

Binomial Expressions

Binomial expressions are algebraic expressions that contain exactly two terms, which are often separated by a plus (\(+\)) or a minus (\(-\)) sign. They are the basic building blocks of much more complex algebraic structures and appear frequently in polynomial factoring.

In the difference of two squares method, each term of a binomial expression becomes component \(a\) or \(b\) in the \(a^2 - b^2\) structure. Following the steps outlined in our example, the expression \(25x^{4}-16y^{6}\) is factored into the product of two binomials, \(5x^2 + 4y^3\) and \(5x^2 - 4y^3\), by recognizing that \(25x^{4}\) and \(16y^{6}\) are perfect squares. By being able to rewrite polynomials as binomial expressions, students can simplify the problem to find solutions more easily and develop a deeper understanding of how algebraic expressions are interconnected.