Problem 29 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. $$2 v^{8}-18 v^{7}-24 v^{6}+2 v^{5}$$

Step-by-Step Solution

Answer

The factored polynomial with the greatest common factor factored out is \(v^5(2 v^3 - 18 v^2 - 24 v + 2)\).

Step 1: Finding the GCF of coefficients

First, we need to find the GCF of the coefficients of the polynomial. The coefficients are 2, -18, -24, and 2. The GCF of these numbers is 2.

Step 2: Finding the GCF of variable terms

Next, we need to find the GCF of the variable terms, which are \(v^8\), \(v^7\), \(v^6\), and \(v^5\). The GCF of these terms is \(v^5\) since it is the lowest power of v shared by all terms.

Step 3: Factoring out the GCF

Now that we know the GCF of both coefficients and variable terms, we can factor out the GCF 2\(v^5\) from each term of the polynomial. $$2 v^{8}-18 v^{7}-24 v^{6}+2 v^{5} = v^5(2 v^3 - 18 v^2 - 24 v + 2)$$

Step 4: Check the answer

To check our answer, we can distribute the GCF back to each term and check if we get the original polynomial. $$v^5(2 v^3 - 18 v^2 - 24 v + 2) = 2 v^{8}-18 v^{7}-24 v^{6}+2 v^{5}$$ Since the factored form simplifies back to the original polynomial, our answer is correct. The factored polynomial is: $$v^5(2 v^3 - 18 v^2 - 24 v + 2)$$

Key Concepts

Greatest Common Factor

Factoring polynomials often starts with determining the greatest common factor (GCF). The GCF is the largest factor that two or more numbers share. In polynomials, this process includes both the numerical coefficients and the variables.
The GCF of numerical coefficients is determined by identifying the largest number that divides all of them without leaving a remainder. For example, in the coefficients 2, -18, -24, and 2, the GCF is 2. This is because 2 is the largest number that can divide each of these coefficients evenly.
This step is crucial because factoring out the GCF simplifies the polynomial, making further manipulation easier. Once the GCF of coefficients is found, you'll then focus on the variable terms, which involve determining the lowest power of the variable common in all terms.

Polynomial Coefficients

In a polynomial, coefficients play a vital role. They are the numbers in front of the variable terms, like the 2 in 2\(v^8\) or -18 in -18\(v^7\). Coefficients affect the size of each term and the polynomial as a whole.
Understanding coefficients helps you manage polynomial equations, especially when factoring.

• First, list all coefficients from the terms, in our case: 2, -18, -24, and 2.
• Then identify their greatest common factor, which helps in simplifying the expression.

Determining the GCF of these coefficients is akin to finding the largest number that can evenly divide all the listed numbers. By dividing each term's coefficient by the GCF, you ensure the polynomial is factored down efficiently, making it simpler to work with.

Variable Terms

Variable terms in a polynomial are the components involving letters that represent numbers, marked by powers or exponents. For example, in the polynomial given, the variable terms are \(v^8\), \(v^7\), \(v^6\), and \(v^5\).
Finding the GCF of these terms involves focusing on the variable's exponents. The goal is to find the term with the smallest exponent common to all, which in this instance is \(v^5\).

• Look at the powers: 8, 7, 6, and 5.
• The smallest is 5, so \(v^5\) is the GCF for the variable terms.

Factoring this GCF simplifies the polynomial, clearing a path for easier solving and manipulation. Understanding the exponents of variable terms is essential for effective polynomial factoring.