Problem 25 - Introductory Algebra for College Students

Question

Factor each difference of two squares. $$16 x^{4}-81$$

Step-by-Step Solution

Answer

The factorization of the expression \(16x^{4} - 81\) is \((4x^{2} + 9)(2x + 3)(2x - 3)\)

Step 1: Identify squares in the expression

The given expression is \(16x^{4} - 81\). Here, the squares are \((4x^{2})^{2}\) and \((9)^{2}\). Hence the expression is of the form \(a^{2} - b^{2}\) where \(a = 4x^{2}\) and \(b = 9\).

Step 2: Apply the difference of squares formula

We apply the difference of squares formula, \(a^{2} - b^{2} = (a+b)(a-b)\), to the expression \(16x^{4} - 81\), we get \((4x^{2} + 9)(4x^{2} - 9)\).

Step 3: Further factor the difference of squares

Here, we observe that \(4x^{2} - 9\) is also a difference of squares where \(a = 2x\) and \(b = 3\). Applying the difference of squares formula again to this term, we get \((2x + 3)(2x - 3)\). Hence, the expression \(16x^{4} - 81\) can be factorized to \((4x^{2} + 9)(2x + 3)(2x - 3)\).

Key Concepts

Factoring Polynomials

Factoring polynomials means breaking them down into simpler parts, called factors, that when multiplied together recreate the original expression. Consider it like splitting a large number into its smaller divisors.

A key method in factoring is identifying patterns, such as the difference of squares, which simplifies the process. A difference of squares is uniquely easy to spot. It takes the form of \(a^2 - b^2\). These expressions are special because they can be quickly split into \((a+b)(a-b)\). This shortcut makes working with polynomials efficient once you get the hang of spotting this form!

In our specific problem, \(16x^4 - 81\) is a difference of squares. Why? Because \(16x^4\) is \((4x^2)^2\) and \(81\) is \(9^2\). Thus, applying the formula gives us \((4x^2 + 9)(4x^2 - 9)\).

Sometimes, you'll see that one part of the result can be further factored. For example, \(4x^2 - 9\) is also a difference of squares and breaks down further with the same technique: \((2x+3)(2x-3)\). Using methods like this helps you tackle more advanced problems in algebra by starting with simpler steps.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at times, operators. They form the building blocks of algebra.

Understanding these expressions involves recognizing parts like terms, coefficients, and like terms. In \(16x^4 - 81\), \(16x^4\) and \(-81\) are two terms: one is a power of \(x\) and the other is a constant. Each piece provides clues on how to simplify or factorize the expression.

We recognize that algebraic expressions can often be simplified using special patterns or identities. One popular identity is the difference of squares. It’s crucial for simplifying expressions like \(16x^4 - 81\).

These expressions can change form without changing their core value using factorization or expansion, depending on what you need. Mastery of these techniques can greatly ease the journey through high school algebra, making difficult problems more digestible.

High School Algebra

High school algebra is where you dive deeper into concepts like factoring, solving equations, and working with algebraic expressions, all critical fundamentals of mathematics.

You learn through structured problems, often involving polynomials, equations, and rational expressions. Factoring equips you with skills to simplify complex problems. For example, by mastering the difference of squares today, you’ll have a powerful tool to deal with similar problems in the future.

A good grasp of these foundational skills propels you into more advanced math classes. It lays a solid base for calculus and other higher math, where these simpler concepts are the underpinnings for more complex theories. Plus, the problem-solving skills you hone here are invaluable beyond school, in real-world situations that require logical thinking and analysis.

High school algebra encourages finding patterns, solving progressively challenging equations, and expanding your analytical thinking. With each problem, you're not just finding an answer but gaining tools to unlock future mathematical puzzles.