Problem 39 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ \left(1-y^{5}\right)\left(1+y^{5}\right) $$

Step-by-Step Solution

Answer

The product of \((1 - y^{5}) \cdot (1 + y^{5})\) is \(1 - y^{10}\).

Step 1: Identify the Pattern

In the expression \((1 - y^{5}) \cdot (1 + y^{5})\), we can see that it takes the form of \((a - b) \cdot (a + b)\), where \(a = 1\) and \(b = y^5\). This is the difference of squares pattern.

Step 2: Apply the Pattern

This pattern states that \((a - b)(a + b) = a^{2} - b^{2}\). When we substitute our \(a = 1\) and \(b = y^{5}\) into the formula, we get \(1^2 - (y^5)^2\).

Step 3: Simplify the Expression

\(1^2 - (y^5)^2\) simplifies to \(1 - y^{10}\). This is the product of the given binomials.

Key Concepts

Polynomial Multiplication

When multiplying polynomials, it's like putting together pieces of a puzzle. Each part needs to match up just right to get the full picture. In the math world, this means combining like terms and applying the distributive property to multiply each term in one polynomial by every term in the other.

For instance, if we're working with \(a + b\) and \(c + d\), we'd distribute to get \(ac + ad + bc + bd\). But some puzzles have shortcuts based on patterns, like the exercise with \(1 - y^{5}\) and \(1 + y^{5}\). This is what we call a difference of squares, a special pattern in polynomial multiplication where you end up with a simpler product: \(a^{2} - b^{2}\).

Understanding this shortcut means less work and fewer chances for mistakes, especially when dealing with higher powers or more complex terms.

Factoring Binomials

Factoring is like un-doing multiplication. It's breaking down an expression into the products that would multiply to make it. For binomials, especially those that follow special patterns, factoring can be a swift process.

One of these patterns is the difference of squares, like our \(1 - y^{10}\) here. If we needed to go in reverse, we would recognize this as \(a^2 - b^2\) and factor it back into \(a + b\) times \(a - b\).

Understanding how to factor binomials and recognizing these patterns isn't just a party trick for algebra enthusiasts—it's a fundamental skill that helps with simplifying expressions and solving equations. When you factor binomials efficiently, you pave the way for clearer solutions in more complex algebraic equations.

Algebraic Patterns

Algebraic patterns, like the difference of squares in the exercise, help us make sense of, and work with, algebraic expressions much more easily. They're recurring themes in the vast story of mathematics, giving us a set of tools to handle otherwise daunting tasks.

The difference of squares pattern, identified by an expression of the form \(a^2 - b^2\), is a result of multiplying two specific binomials: \(a + b\) and \(a - b\). Spotting this and other patterns not only saves time but also builds a deeper understanding of how algebra works.

By recognizing these patterns, you can often bypass long, tediously detailed calculations and move straight into simplification. Just like knowing the shortcut home avoids a traffic jam, knowing algebraic patterns, provides a direct route to the answer, making your mathematical journey smoother and quicker.