Problem 46 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ \frac{2}{5} y^{7}-\frac{4}{5} y^{5}+\frac{3}{5} y^{2}-\frac{2}{5} y $$

Step-by-Step Solution

Answer

The factored polynomial is \(\frac{1}{5}y (2y^6 - 4y^4 + 3y - 2)\).

Step 1: Identify the Common Factor in Coefficients

Examine the coefficients of each term in the polynomial: \(\frac{2}{5}, -\frac{4}{5}, \frac{3}{5}, -\frac{2}{5}\). The common factor for these coefficients is \(\frac{1}{5}\).

Step 2: Identify the Common Factor in Variables

Examine the variable part of each term: \(y^7, y^5, y^2, y^1\). The lowest power of \(y\) present in all terms is \(y^1\), or simply \(y\). Thus, \(y\) is the common factor in variables.

Step 3: Factor out the GCF

Combine the common factors identified from the coefficients and the variables. The overall GCF is \(\frac{1}{5}y\). Factor out \(\frac{1}{5}y\) from each term in the polynomial:\[\frac{1}{5}y (2y^6 - 4y^4 + 3y - 2)\]

Step 4: Reassemble the Factored Polynomial

After factoring out \(\frac{1}{5}y\), the expression inside the parentheses is what remains from each term after dividing by the GCF. This results in \(2y^6 - 4y^4 + 3y - 2\) inside the parentheses.

Key Concepts

Factoring Polynomials

Factoring polynomials is a key skill in algebra that involves breaking down a complex polynomial into simpler components, known as factors. To do this, we often identify the greatest common factor (GCF) among the terms and factor it out. Factoring makes it easier to solve polynomial equations and simplify expressions.

In our example, the polynomial \(\frac{2}{5} y^{7}-\frac{4}{5} y^{5}+\frac{3}{5} y^{2}-\frac{2}{5} y\) can be factored by first finding and extracting the GCF. This involves:

• Determining the GCF of the coefficients
• Finding the GCF of the variable terms

By factoring out the GCF, \(\frac{1}{5}y\), we simplify the polynomial into its factored form: \(\frac{1}{5}y(2y^6 - 4y^4 + 3y - 2)\).

This new expression is easier to manipulate or analyze because it is reduced to a simpler form where every operation involving the polynomial first considers the GCF.

Polynomial Terms

A polynomial term is made up of a coefficient, a variable, and an exponent which shows the power to which the variable is raised. In the context of our given polynomial, each term such as \(\frac{2}{5}y^7\) consists of a coefficient \(\frac{2}{5}\), a variable \(y\), and an exponent \(7\).

This pattern repeats in each polynomial term:

• Coefficient: A fractional number like \(\frac{3}{5}\)
• Variable: The same base variable \(y\)
• Exponent: Powers such as \(2\), \(5\), or \(7\)

Recognizing these components in each term helps in identifying common factors across a polynomial. Once found, they can be used to simplify the entire expression, as demonstrated by our initial extraction of the GCF.

Algebraic Expressions

Algebraic expressions encompass polynomial expressions that include numbers, variables, and arithmetic operations like addition or subtraction. They form the building blocks of algebra and can represent a wide range of quantities.

Consider the algebraic expression \(\frac{2}{5}y^{7} - \frac{4}{5}y^{5} + \frac{3}{5}y^{2} - \frac{2}{5}y\). This particular expression comprises multiple polynomial terms linked together. Each term contributes a piece to the overall expression, so handling them together often requires factoring to find their common characteristics.

Evaluating such expressions, especially involving factoring, helps in simplifying complex equations, solving for unknowns, or even translating real-world problems into manageable mathematical terms. Thus, understanding and manipulating algebraic expressions are key to mastering algebra.