Problem 2 - Elementary and Intermediate Algebra

Question

Fill in the blanks. When the polynomial \(8 x^{3}+125\) is written as \((2 x)^{3}+(5)^{3},\) we see that it is the sum of two ________.

Step-by-Step Solution

Answer

The polynomial is the sum of two cubes.

Step 1: Identify the Form of the Polynomial

The polynomial given is \(8x^{3} + 125\). Recognize that this polynomial can be rewritten as \((2x)^{3} + 5^{3}\). This indicates the polynomial is expressed in the form of \(a^{3} + b^{3}\), where \(a = 2x\) and \(b = 5\).

Step 2: Recognize the Structure of a Sum of Cubes

Understand that the form \(a^{3} + b^{3}\) is known as the sum of two cubes. This is because it represents two terms, each raised to the power of three.

Key Concepts

Sum of Cubes

In algebra, the term "sum of cubes" refers to an expression where two cube numbers are added together. Specifically, it takes the form \(a^3 + b^3\). This is distinguished from a single cube, as it involves a combination of two separate cubic terms.
Recognizing a sum of cubes is crucial because it often appears in polynomial problems. For example, in the polynomial \(8x^3 + 125\), it can be rewritten as \((2x)^3 + 5^3\). Here, \( (2x)^3 \) and \( 5^3 \) are two cubes, thus forming a sum of cubes.
This structure is vital in simplifying or factoring polynomials, as it provides a method to break down complex expressions. It allows you to utilize special factorization formulas to further resolve the equation.

Cubic Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. Cubic polynomials are a specific kind where the highest degree of the variable is three.
For example, in the polynomial \(8x^3 + 125\), the term \(8x^3\) identifies it as cubic, indicating that the variable \(x\) is raised to the third power.
Cubic polynomials can be involved, as they may have up to three real roots, and their graphs can take various shapes involving up to two turning points. The ability to manipulate them, such as rewriting a cubic polynomial in terms of a sum of cubes, can aid in simplifying the polynomial and making it more manageable. Understanding cubic structures is essential for solving higher-order algebraic expressions.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components, or factors, that when multiplied together will yield the original polynomial. It is a critical skill to master because it simplifies complex expressions and makes solving equations more feasible.
For the sum of cubes, such as \(a^3 + b^3\), there is a specific factoring formula which is:

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

This formula provides a method to rewrite a sum of cubes as a product of a binomial and a trinomial.
Take the expression \((2x)^3 + 5^3\). By identifying \(a = 2x\) and \(b = 5\), you can apply the sum of cubes formula to factor it. Factoring allows not only for simplification of expressions but also aids in solving equations where the variable exists in a cubic relationship.