Problem 53 - Calculus: One and Several Variables

Question

Write the expression in factored form. \(8 x^{6}+64\).

Step-by-Step Solution

Answer

After factoring, the expression \(8 x^{6}+64\) can be written in its simplest form as \(8(x^2 + 2)(x^4 - 2x^2+ 4)\)

Step 1: Identify the common factor

Firstly, look for any terms they have in common which could be factorized out. Here it appears 8 could be taken out, leaving \(8(x^{6}+8)\).

Step 2: Rewrite the expression by recognizing the cubes

Secondly, rewrite the expression by recognizing the cubes. The given expression can be rewritten as \(8(x^{2})^{3}+2^{3}\). The form is now \(a^{3}+b^{3}\).

Step 3: Apply the identity for the sum of cubes

Finally, apply the identity for the sum of cubes: \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\). Here, a is \(x^{2}\) and b is 2.: \((x^2 +2)((x^{2})^{2} - 2x^{2} + 2^{2})\). After simplifying, you get \((x^2 + 2)(x^4 - 2x^2+ 4)\)

Key Concepts

Sum of Cubes

The "sum of cubes" is a special way to factor polynomials that have the form \( a^3 + b^3 \). In this case, each term is a perfect cube, meaning they can be expressed as the cube of some term. To apply the sum of cubes formula, identify your "\( a \)" and "\( b \)" in the expression. The formula is:

• \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)

This formula provides a neat shortcut for factoring such expressions, saving you from doing laborious polynomial division. In our example, we identified \( a \) as \( x^2 \) and \( b \) as 2, allowing us to factor the expression into a simple product of binomials and trinomials.
Understanding this formula is crucial in higher-level algebra as these patterns frequently appear, especially in polynomial equations.

Algebraic Identities

Algebraic identities are mathematical statements that hold true for all values of the variables involved. These identities help in simplifying and factoring complex expressions. For instance, the "sum of cubes" is a widely recognized identity. Learning identities like these is fundamental in algebra.
It's important to remember that similar algebraic identities exist for different forms, such as "difference of squares" or "sum and difference of cubes." Each has its own distinct form and usage:

• Difference of Squares: \( a^2 - b^2 = (a - b)(a + b) \)
• Difference of Cubes: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)

By harnessing these identities, you can quickly solve problems that might otherwise seem daunting. Recognizing patterns and knowing which identities to apply are skills that develop with practice and experience in high school mathematics.

High School Mathematics

High school mathematics lays the foundation for understanding various mathematical principles, including factoring polynomials. Algebraic techniques, like recognizing patterns such as the sum of cubes, are key elements taught at this level. These skills are essential not only for mathematics but also for fields like physics and engineering.
In high school algebra:

• You learn to factor different types of polynomials.
• There is an emphasis on practice to solidify understanding.
• Exercises typically start with identifying simple expressions, and then progress to using algebraic identities.

By mastering these concepts, you're equipping yourself with tools necessary for advanced studies. Solving polynomial equations is not only a high school milestone but also a stepping stone for understanding calculus and other advanced mathematics topics later in education.