Problem 84 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ x^{3}+64 $$

Step-by-Step Solution

Answer

\((x + 4)(x^2 - 4x + 16)\)

Step 1: Identify the Type of Polynomial

The polynomial given is \(x^3 + 64\). Notice that it is a sum of cubes since \(64 = 4^3\). The polynomial fits the form \(a^3 + b^3\), where \(a = x\) and \(b = 4\).

Step 2: Recall the Sum of Cubes Formula

The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). We will use this formula to factor the given polynomial \(x^3 + 64\).

Step 3: Apply the Formula to Factor the Polynomial

Using \(a = x\) and \(b = 4\), substitute into the sum of cubes formula: \(x^3 + 64 = (x + 4)(x^2 - 4x + 16)\). This is the factored form of the polynomial.

Key Concepts

Sum of Cubes Formula

Understanding the sum of cubes formula is essential when dealing with specific types of polynomials like the one in the exercise. The sum of cubes formula helps in factoring expressions such as \(a^3 + b^3\). This formula is given by:

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

This is a powerful tool because it simplifies the polynomial into a product of two expressions, which are easier to handle.
For any sum of cubes, simply identify your \(a\) and \(b\) values. Then, plug them into the formula. In the exercise problem, \(x^3 + 64\) is written as \(x^3 + 4^3\). Thus \(a = x\) and \(b = 4\). Applying the formula gives the factored form \((x + 4)(x^2 - 4x + 16)\).
This method is not only straightforward but also consistent across different sums of cubes expressions, making it a reliable strategy in algebra.

Algebra

Algebra involves the manipulation and understanding of symbols and letters to solve equations. It's a broad area in mathematics, but its core lies in using principles and operations to reason about numbers and quantities. In this context, the exercise on factoring polynomials is a typical algebra problem involving higher-degree polynomial expressions.
In algebra, to handle polynomials, we often look for patterns or known formulas—like the sum of cubes formula. Recognizing structures like \(x^3 + 4^3\) allows us to apply efficient techniques for simplification. These skills build upon understanding basic operations and extending them to work with more complex polynomial expressions.
By learning to factor polynomials, you gain valuable insights into how numbers relate under operations of multiplication and addition. This deepens your ability not only to perform computations but also to solve problems creatively.

Polynomial Expressions

Polynomial expressions are sums of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient. The expression provided, \(x^3 + 64\), includes two terms, and it's a simple example of how polynomial expressions work.

• Every polynomial can be classified based on its degree, which is determined by the highest power of the variable.
• The degree of \(x^3 + 64\) is 3, as it has the highest power of 3.

Understanding polynomial expressions involves recognizing their structure and knowing how to manipulate them through operations like addition, subtraction, multiplication, and division.
Factoring is one way to work with polynomials, and it simplifies them into products of simpler expressions such as binomials or trinomials, like we did with the sum of cubes formula. Whether working with single-variable or multi-variable polynomials, realizing the power of factoring can be a key skill in many areas of math and applied sciences.