Problem 107 - College Algebra

Question

Factor completely. $$ y^{7}+y $$

Step-by-Step Solution

Answer

The factored form of the given expression is \(y(y^{6} + 1)\)

Step 1: Identify common factors

First, look for any common factors present in all terms in the equation. Here, both \(y^{7}\) and \(y\) have \(y\) as a common factor.

Step 2: Factor out the common factor

Next, factor out the common factor from the terms. It means we write the expression as \(y(y^{6} + 1)\).

Step 3: Check if the expression can be factored further

Then, see if the remaining expression \(y^{6} + 1\) can be factored any further. In this case, it cannot be factored further because it is the addition of two terms that share no further common factors.

Key Concepts

Understanding Common Factors

When we talk about common factors in the context of algebra, we refer to the numbers, variables, or expressions that are present in all the terms of an algebraic expression. Spotting common factors is an essential skill, as it's the first step in simplifying complex algebraic expressions and it paves the way for further factorization.

For example, consider the algebraic expression:
\( y^7 + y \).
Both terms – \( y^7 \) and \( y \) – have a \( y \) in common. Factoring out this common factor gives us:
\[ y(y^6 + 1) \].
The expression is now simpler and easier to handle. Always remember, a common factor can be a number (e.g., 2 in 2x and 2y), a variable (e.g., x in x^2 and x^3), or even a combination of both.

Algebraic Expressions Simplification

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operations (like addition, subtraction, multiplication, and division). The goal when working with algebraic expressions is often to simplify them or rewrite them in a more useful form by factoring.

The expression from our example, \( y^7 + y \), looks simple, but it hides a common factor. By factoring out \( y \), the expression becomes more manageable. Understanding how to manipulate these expressions is crucial for solving a variety of problems in algebra. That’s why identifying common factors and using them to simplify expressions is a key tactic to master.

Applying the Factor Theorem

The Factor Theorem is a powerful tool in algebra that connects zeros of a polynomial with its factors. It states that if a polynomial \( f(x) \) has a zero at \( x = c \), then \( x - c \) is a factor of \( f(x) \). Conversely, if \( x - c \) is a factor of \( f(x) \), then \( c \) is a zero of the polynomial.

Although the Factor Theorem is incredibly useful, it relies on the polynomial having zeros that can be readily identified. In our example, the simplified expression \( y(y^6 + 1) \) doesn't readily allow us to apply the Factor Theorem since it doesn't have obvious zeros. If we had a case where the polynomial had a form like \( y^2 - 1 \), we could apply the theorem to factor it into \( (y + 1)(y - 1) \), because both 1 and -1 are zeros of the polynomial \( y^2 - 1 \).