Problem 84 - Introductory Algebra for College Students

Question

Factor completely: \(x^{4}+2 x^{3}-3 x-6 .\) (Section 6.1 Example 8).

Step-by-Step Solution

Answer

The complete factorization of the polynomial \(x^{4} + 2 x^{3} - 3 x - 6\) is \(3x(x^{3} - 3)\).

Step 1: Rearrange polynomial

Rearrange the polynomial as: \(x^{4}+2 x^{3} - 3x - 6 = x^{4} - 3x + 2 x^{3} - 6\) .

Step 2: Regroup terms

Regroup the terms as: \(x^{4} - 3x + 2 x^{3} - 6 = x^{4}+ 2 x^{3} - 3x - 6 = (x^{4} - 3x) + (2 x^{3} - 6)\).

Step 3: Identify common factors

Identify common factors within the groups as: \(x^{4} - 3x + 2 x^{3} - 6 = (x^{4} - 3x) + (2 x^{3} - 6)= x(x^{3} - 3) + 2x(x^{3} - 3)\).

Step 4: Factorize the polynomial completely

Factorize completely using common binomial as: \(x^{4} - 3x + 2 x^{3} - 6 = x(x^{3} - 3) + 2x(x^{3} - 3) = (x + 2x)(x^{3} - 3) = 3x(x^{3} - 3)\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can look quite complex, with many terms connected by plus or minus signs. Each term in an algebraic expression is called a "monomial," made up of a coefficient (a number), a variable (like \(x\) or \(y\)), or a combination of both. A single-term algebraic expression could be something as simple as \(5x\), while a multi-term expression could be like \(x^4 + 2x^3 - 3x - 6\).

An important aspect of algebraic expressions is simplifying them through various methods. Simplification often focuses on reducing expressions by combining like terms—terms with the same variable raised to the same power. Understanding how to handle algebraic expressions is crucial for higher-level math topics, such as calculus or advanced algebra. It helps us generalize and model real-world situations into mathematical equations.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler "factors" that, when multiplied together, give back the original polynomial. The goal is to express a complex polynomial as a product of its factors. This skill is valuable because it helps in solving equations, simplifying expressions, and understanding mathematical relationships.

Let's consider the polynomial \(x^4 + 2x^3 - 3x - 6\). The purpose is to see if it can be split into smaller, more manageable polynomials. By rearranging terms and grouping them smartly, factorization simplifies calculations and is often employed in solving polynomial equations.

Factorization is a foundational algebraic technique. Familiar methods include the greatest common factor (GCF), grouping, or using formulas for special polynomials such as the difference of squares. Each has its place, depending on the structure of the polynomial in question.

Common Factors

Finding common factors within polynomial terms is a crucial step in the factorization process. A common factor is a number or expression that divides each term of the polynomial without leaving a remainder. By identifying these, you can simplify expressions and make further factorization possible.

Consider the expression \(x^4 - 3x + 2x^3 - 6\). This polynomial can be regrouped into \((x^4 - 3x) + (2x^3 - 6)\). Here, \(x\) is a common factor in both terms of the first group. By factoring \(x\) out of both sections, the expression simplifies to \(x(x^3 - 3) + 2x(x^3 - 3)\). You can then factor out the common binomial \((x^3 - 3)\) to further simplify the expression.

Identifying common factors requires keen observation and practice. Skillfully doing so can significantly ease solving polynomials and improve algebraic manipulation skills.