Problem 21 - Elementary and Intermediate Algebra

Question

Factor. $$ n^{3}+512 $$

Step-by-Step Solution

Answer

The factorization is \((n + 8)(n^2 - 8n + 64)\).

Step 1: Recognize the Sum of Cubes

The expression \(n^3 + 512\) can be written as the sum of two cubes. Notice that \(512\) is a perfect cube, specifically \(512 = 8^3\). Therefore, the expression can be rewritten as \(n^3 + (8)^3\).

Step 2: Apply the Sum of Cubes Formula

The sum of cubes formula is: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). In our problem, we have \(a = n\) and \(b = 8\). Substitute \(a\) and \(b\) into the formula to get:\[(n + 8)(n^2 - 8n + 64)\].

Step 3: Simplify the Expression

Multiply to check if each part expands back to the original expression:1. First part: \((n + 8)\) 2. Second part: \((n^2 - 8n + 64)\)Use distribution to verify it's correct:\[(n+8)(n^2) = n^3 + 8n^2\] \[(n+8)(-8n) = -8n^2 - 64n\]\[(n+8)(64) = 64n + 512\]Bringing all together gives: \[n^3 + 8n^2 - 8n^2 - 64n + 64n + 512 = n^3 + 512\].The factorization is correct.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental aspect of algebra that involves breaking down a complex expression into simpler parts or products. It is like solving a puzzle, where the end goal is to express the polynomial as the product of its factors.
When you factor, you look for expressions that multiply together to give the original polynomial. This can simplify computations and help solve equations more easily.
There are different methods of factoring polynomials, including:

• Common Factor: Look for common factors in all terms.
• Difference of Squares: Used when two squares are subtracted.
• Sum or Difference of Cubes: Used when dealing with cubic terms.

In our exercise, we specifically use the sum of cubes method to factor the expression into simpler parts. This method requires us to recognize the form \(a^3 + b^3\) and apply the corresponding factoring formula.
By mastering different factoring techniques, you can handle a wide range of algebraic expressions more effectively.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operational symbols. It can represent real-world situations or purely theoretical concepts, allowing us to perform calculations and solve problems.
Understanding the components of an algebraic expression is crucial:

• Variables: Symbols that represent unknown values, such as \(n\) in our exercise.
• Constants: Specific numerical values, like \(512\) which is a perfect cube.
• Coefficients: Numbers multiplying the variables, though in \(n^3 + 512\) the coefficients are implicitly \(1\) for \(n^3\).
• Operators: Symbols that indicate operations, such as addition or subtraction.

By rearranging and manipulating these components using algebraic rules, you can simplify expressions and solve equations. This process is the backbone of algebra and critical for handling more complex mathematical problems effectively.

Perfect Cubes

Perfect cubes are special numbers that can be expressed as the cube of an integer. Recognizing these numbers is helpful when dealing with polynomial expressions like sums or differences of cubes.
In our problem \(n^3 + 512\), recognizing \(512\) as a perfect cube, specifically \(8^3\), allows us to apply the sum of cubes method successfully.
Some common perfect cubes include:

• \(1^3 = 1\)
• \(2^3 = 8\)
• \(3^3 = 27\)
• \(4^3 = 64\)
• \(5^3 = 125\)
• and so on.

Identifying perfect cubes quickly helps factor expressions efficiently by applying specific algebraic formulas. It is an essential skill when working with cubic equations and understanding their roots and behavior.