Problem 101 - Introductory Algebra for College Students

Question

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) $$0.25 x-x^{3}$$

Step-by-Step Solution

Answer

\(0.25x - x^{3} = x(0.25 - x^{2})\)

Step 1: Identify common factor

First things first, we will look at the two terms in the expression, \(0.25x\) and \(-x^{3}\). We need to find if there is a common factor, which in this case is \(x\).

Step 2: Factor out the common factor

The next step is to divide each of the terms by the common factor, \(x\). Doing that, we would have \((0.25 - x^{2})x\). This results in the factored form of the equation.

Key Concepts

Common Factor

Finding the common factor in polynomial expressions is a fundamental step in factoring completely. In the given exercise, we have two terms: \(0.25x\) and \(-x^3\). The task is to find a common factor that exists in both terms. Here, the common factor is \(x\).

What you do next is divide each term by the common factor. This step simplifies the expression, making it easier to factor. When you remove \(x\) from each term, you are left with \((0.25 - x^2)x\). This is a crucial move because it reduces complexity in polynomial expressions and allows further manipulations, if necessary.

Remember:

• Identify terms first.
• Search for a common variable or number in all terms.
• Factor it out by dividing each term by this factor.

Rational Numbers

Rational numbers are numbers that can be expressed as fractions where both the numerator and denominator are integers. They are essential when factoring polynomial expressions as they affect both the coefficients and the overall expression.

In our exercise, the coefficient \(0.25\) is a rational number, as it can be written as \(\frac{1}{4}\). This knowledge is useful when factoring because recognizing coefficients as rational numbers can help simplify and factor the expression efficiently.

The ability to recognize rational numbers aids in maintaining the accuracy of your transformations when working through polynomial expressions:

• Recognize rational numbers by identifying how coefficients can convert into fractions.
• Use these conversions to simplify and factor expressions effectively.
• Rational numbers form a vital part of understanding the mathematical relationships in polynomials.

Polynomial Expressions

Understanding polynomial expressions is key to effectively factoring them. A polynomial is a mathematical expression consisting of variables (like \(x\)) and coefficients, which are often rational numbers.

In the given expression \(0.25x-x^3\), we have a polynomial with two terms, each with different powers of \(x\). Structurally, this polynomial is arranged in decreasing order of the power of \(x\). Recognizing this pattern can assist you when you're next tasked with factoring.

Important points about polynomial expressions include:

• Polynomials are often written with terms in descending order by their degree.
• Each term consists of a base and an exponent coupled with a coefficient.
• Factoring often aims to express these expressions in a simpler and often more useful form, such as a product of factors.

Whether you're dealing with monomials, binomials, or higher-degree polynomials, understanding their structure helps in simplifying, expanding, and factoring them efficiently.