Problem 100 - Introductory Algebra for College Students

Question

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) $$y^{4}-\frac{y}{8}$$

Step-by-Step Solution

Answer

The completely factored form of \(y^{4} - \frac{y}{8}\) is \(y( y^{3} - \frac{1}{8})\).

Step 1: Identify the Common Factor

First, determine if there is a common factor in all terms. In our case, the common factor is \(y\).

Step 2: Factor out the Common Factor

Now, factor out the common factor. This will result in: \(y( y^{3} - \frac{1}{8})\)

Step 3: Look for Additional Factors

Now, consider the second expression which is a simple polynomial. This cannot be further factored since it does not have any common factors. Hence, our final complete factored form is \(y( y^{3} - \frac{1}{8})\).

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. They are fundamental in mathematics because they fill the gap between integers and can represent values such as \( \frac{1}{2} \) or \( -\frac{3}{4} \). In the expression given, \( y^{4}-\frac{y}{8} \), the term \( \frac{1}{8} \) is a rational number because it can be represented as a fraction of two integers: 1 and 8.

Understanding rational numbers is essential when dealing with polynomials because sometimes coefficients of polynomials can be rational. This can affect how polynomials are factored. For example, rational expressions must be simplified fully in their fractional parts when factoring polynomials.

Simply put, whenever you see a rational number in a polynomial, it should be treated with care to maintain the integrity of the factorization process.

Common Factors

Common factors in the context of polynomials are terms that are present in all parts of the polynomial expression. Identifying and factoring out common factors is often the first step in the process of simplifying or factoring expressions. For the polynomial expression \( y^{4} - \frac{y}{8} \), the term \( y \) appears in both parts. Thus, \( y \) is identified as a common factor.

When you factor out the common factor \( y \), what you're doing is reducing each term by this factor. So, the expression \( y^{4} - \frac{y}{8} \) becomes \( y(y^{3} - \frac{1}{8}) \). This simplifies the expression while maintaining its original meaning.

By understanding common factors, you will be able to simplify complex polynomial expressions into simpler ones, making further calculations much easier and providing clarity to the problem at hand.

Polynomial Expression

A polynomial expression comprises one or more terms which involve variables raised to whole number powers. Polynomials are classified based on the number of terms: monomials, binomials, trinomials, etc. For our exercise, \( y^{4} - \frac{y}{8} \) is a binomial as it has two main terms.

Each term in a polynomial can include coefficients (which might be rational numbers), variables, and exponents. In the provided expression, \( y^{4} \) and \( \frac{y}{8} \) are the two terms. The operation between the terms is subtraction. When factoring polynomial expressions, especially binomials, it's crucial to identify any possible common factors in both terms, as we did with the \( y \) in this exercise.

Mastering the basics of polynomial expressions will enable you to factor and manipulate these mathematical entities effectively, making complex algebraic problems more manageable.