Problem 102 - Introductory Algebra for College Students

Question

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) $$0.64 x-x^{3}$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(0.64x - x^{3}\) is \(x(0.64 - x^{2})\).

Step 1: Identify the Greatest Common Factor (GCF)

In this polynomial, the GCF that can be factored out from both terms is \(x\). This is due to both terms, \(0.64x\) and \(-x^{3}\), sharing \(x\) as a common factor.

Step 2: Factor out the GCF from both terms

By factoring out the GCF \(x\), we rewrite the polynomial as \(x(0.64 - x^{2})\). Here, we've simply divided each term in the original polynomial by \(x\), giving us two simpler expressions.

Step 3: Simplify and Write Final Answer

The factored polynomial does not simplify further. Thus, the completely factored form of \(0.64x - x^{3}\) is \(x(0.64 - x^{2})\).

Key Concepts

Greatest Common Factor

Understanding the greatest common factor (GCF) is essential when factoring polynomials. The GCF of a set of terms is the largest expression that divides all the terms without leaving a remainder. In algebra, identifying and factoring out the GCF simplifies expressions and reveals simpler polynomial forms.

For example, consider the polynomial \(0.64x-x^{3}\). The GCF is \(x\), the term that both \(0.64x\) and \(x^{3}\) share. By removing \(x\) from each term, you're essentially dividing both by \(x\), which streamlines the expression to \(x(0.64 - x^{2})\). This process of extraction makes further operations on the polynomial more manageable and is a foundational skill for working with more complex algebraic expressions.

Polynomial Simplification

Simplifying a polynomial involves several techniques, including factoring, combining like terms, and applying the distributive property. The ultimate goal is to reduce the polynomial to its easiest form for computation or further manipulation. Polynomial simplification can reveal properties of the function it represents, like roots and end behavior.

After factoring out the greatest common factor, our goal is to observe if the remaining terms can be simplified further. For the polynomial \(x(0.64 - x^{2}))\), no like terms are present within the parentheses and the terms cannot be factored further algebraically. Therefore, the simplified result remains \(x(0.64 - x^{2}))\). It's crucial to scrutinize the remaining expression carefully, however, because overlooking additional simplification steps could only lead to partial solutions.

Rational Numbers

Rational numbers play a significant role in polynomial expressions. They are numbers that can be expressed as the quotient of two integers, commonly seen in fraction form, such as \(\frac{3}{4}\) or \(\frac{-7}{2}\), including integers and finite decimals which can be re-written as fractions.

In our example, the coefficient \(0.64\) is a rational number since it can be expressed as \(\frac{64}{100}\), which simplifies to \(\frac{16}{25}\). When factoring polynomials, it's important to recognize that coefficients may be rational numbers that can be involved in the factoring process. Acknowledging this could provide insight into possible factorizations, especially when polynomials have more complex structures with rational coefficients.