Problem 107 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The factorization of \(y^{7} + y\) is \(y*(y^{6} + 1)\).

Step 1: Identify Common Factors

In the expression \(y^{7} + y\), both terms share a common factor of y, which can be factored out.

Step 2: Write as a Product of Common Factors

The expression can be written as a product of the common factor and the remaining expression. Hence, \(y^{7} + y\) can be rewritten as \(y*(y^{6} + 1)\). This is the fully factored form of the given expression.

Key Concepts

Understanding Common Factors

In algebra, identifying common factors in polynomial expressions is a crucial step towards simplification and solving equations. A common factor is a term that is present in each term of the polynomial. It's like finding something that all parts of the expression have in common. For example, in the expression y^7 + y, both terms have a y in them, making y a common factor. Factoring this out means we are dividing each term by y and simplifying the expression into a more digestible format.

Finding common factors helps in reducing the complexity of algebraic expressions, which in turn can make other operations like addition, subtraction, or finding which values of variables make the expression equal to zero (solving equations) much easier to perform. This technique is not just a mechanical process but a foundational tool that enhances problem-solving skills and understanding of algebraic structure.

Deciphering Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. To understand an algebraic expression, it's essential to know its components and how they interact. In the given problem, y^7 + y is a simple expression with two terms, connected by addition. Each term is made up of the variable y raised to a power (the exponent).

The power of algebra lies in its ability to generalize mathematical ideas. Instead of working with specific numbers, algebra uses symbols like y to represent any number. This allows us to solve problems in a broad sense and apply our solutions to many different scenarios. Recognizing how terms in an expression relate through common factors is an example of such generalization, opening the door to further manipulation and simplification of complex algebraic expressions.

The Steps of Polynomial Factoring

Factoring polynomials can seem daunting at first, but by breaking down the process into clear steps, it becomes much more approachable. The first step is to look for any common factors across the terms of the polynomial. Once these are identified and factored out, what remains can often be simplified further. With the expression y^7 + y, we started by recognizing that each term contains a y.

Once the common factor y is factored out, we're left with y*(y^6 + 1). Now, this might look like the end, but sometimes, the remaining polynomial y^6 + 1 can be factored further. However, in this case, it's already in its simplest form since there are no more common factors or recognizable patterns (like the difference of squares or perfect square trinomials). Understanding and following these steps are essential for students to build a solid foundation in algebra and tackle more complex problems with confidence.