Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, find the greatest common factor. \(6 y^{4}-2 y^{3}+3 y^{2}-y\)

Step-by-Step Solution

Answer

The greatest common factor is \(y\).

Step 1: Identify the Common Factor in the Coefficients

Look at the coefficients of each term in the polynomial: 6, -2, 3, and -1. The greatest common factor (GCF) of these numbers is 1 since there is no other number greater than 1 that divides all of them evenly.

Step 2: Determine the Lowest Power of the Variable

Identify the variable in each term, which is \(y\) with powers 4, 3, 2, and 1. The lowest power of \(y\) in the polynomial is \(y^1 = y\).

Step 3: Combine the GCF from Coefficients and Variables

To find the GCF of the entire expression, combine the GCF of the coefficients (1) with the lowest power of the variable, \(y\). Thus, the greatest common factor for the polynomial is \(y\).

Key Concepts

Polynomial

A polynomial is a mathematical expression that consists of variables and coefficients, structured together using arithmetic operations such as addition, subtraction, and multiplication. It's a sum of several terms, each term being a product of a coefficient and a variable raised to a power. Polynomials can be simple, having just one term, or complex, with many terms. For example, the polynomial presented in the exercise is made up of four terms:

• 6y⁴
• -2y³
• +3y²
• -y

Each term is expressed with a coefficient and a variable, indicating how these quantities are combined using basic mathematical operations.

Coefficients

Coefficients are the numerical parts of the terms in a polynomial. They are constants that multiply the variables in each term. In the given polynomial, the coefficients are 6, -2, 3, and -1.
These numbers define the scaling factor for the power of the variable in that particular term.

• 6 in 6y⁴
• -2 in -2y³
• 3 in 3y²
• -1 in -y

To determine the greatest common factor of the polynomial's coefficients, we look for the highest number that perfectly divides each coefficient. Here, our step-by-step solution concludes the GCF is 1. This means that apart from 1, no other number can evenly divide them all.

Variables

In a polynomial, variables represent unknown quantities that can change. They appear in the expression accompanied by coefficients, and they can have various powers (or exponents).
The variable in the exercise is 'y'. Variables in polynomials can have different names such as x, t, or z, among others, but in this particular polynomial, we only deal with 'y'. The power to which the variable is raised changes how it impacts the polynomial's overall value. Identifying the consistent variable across terms is crucial for simplifying and factoring polynomials.

Powers

The power of a variable in a polynomial indicates the number of times that variable is multiplied by itself in that term. In the given polynomial, the powers of the variable 'y' range from 1 to 4. In detail:

• In 6y⁴, y is raised to the 4th power, which means it is y multiplied by itself 3 more times (y * y * y * y).
• For -2y³, y is raised to the 3rd power.
• In 3y², y is raised to the 2nd power.
• Finally, in -y, it is raised to the 1st power.

When finding the greatest common factor in a polynomial, it is typically based on the lowest power of the variable present across all terms. Hence, the lowest power is y to the power of 1, which simply gives us the variable itself as part of the GCF.