Problem 2 - Elementary and Intermediate Algebra

Question

The factorization of \(x^{3}+8\) is \((x+2)\left(x^{2}-2 x+4\right) .\) The first factor is a binomial and the second is a _____.

Step-by-Step Solution

Answer

trinomial

Step 1: Identify the Factors

The given expression is a polynomial that has been factored into two parts: \((x+2)\) and \((x^2 - 2x + 4)\). The first part, \((x+2)\), is clearly a binomial since it consists of two terms.

Step 2: Analyze the Second Factor

The second factor is \(x^2 - 2x + 4\). This expression contains three distinct terms: a quadratic term \(x^2\), a linear term \(-2x\), and a constant term \(4\).

Step 3: Classify the Second Factor

An expression with three terms is known as a trinomial. Thus, the second factor \(x^2 - 2x + 4\) is classified as a trinomial.

Key Concepts

Binomial

In mathematics, a binomial is an algebraic expression that contains exactly two terms. These terms are usually joined by a plus "+" or minus "-" sign. Binomials are very common in algebra and may appear in various forms such as

• \(x + y\)
• \(a - b\)
• \(3x + 4\)

A key characteristic of binomials is their simplicity, making them straightforward to factorize and manipulate in equations. When encountering a binomial like \[(x + 2)\]you are dealing with two terms: "\(x\)" and "2". These two pieces form the building blocks for more complex algebraic structures.

Trinomial

A trinomial is an algebraic expression that consists of three terms. An example of a trinomial is

• \(x^2 - 2x + 4\)
• \(a^2 + b^2 + c^2\)
• \(3x^2 + 5x - 7\)

Trinomials are a step up in complexity from binomials due to the additional term. They often appear in the context of quadratic expressions and can be seen in the form \[ax^2 + bx + c\]where

• \(a\) is the coefficient of the squared term
• \(b\) is the coefficient of the linear term
• \(c\) is the constant term.

Understanding trinomials is crucial for solving quadratic equations, as many methods of factoring revolve around breaking down these three-part expressions into simpler binomial products.

Quadratic Expressions

Quadratic expressions are a special type of polynomial that include a squared term as their highest degree. They usually have the form \[ax^2 + bx + c\]where:

• \(a\), \(b\), and \(c\) are constants, and \(a eq 0\)
• \(x\) is the variable.

Quadratic expressions play a key role in algebra because they appear frequently in equations that model real-world phenomena, such as

• Projectile motion
• Optimization problems
• Electrical circuits

There are several methods to tackle quadratic expressions, including factoring them into two binomials, completing the square, or using the quadratic formula. These techniques allow us to solve quadratic equations by making the expression equal to zero and finding the values of \(x\) that satisfy the equation. Recognizing how to manipulate and solve quadratic expressions is essential for advancing in algebra and understanding higher-level mathematics.