Problem 36 - Elementary Algebra

Question

Factor the greatest common factor from each polynomial. \(8 m^{2}-40 m+16\)

Step-by-Step Solution

Answer

8(m^{2} - 5m + 2)

Step 1: Identify the Greatest Common Factor (GCF)

First, determine the GCF of the coefficients and variables in the polynomial. The coefficients are 8, -40, and 16. The GCF of these numbers is 8.

Step 2: Factor Out the GCF

Divide each term in the polynomial by the GCF, which is 8. This gives \[ \frac{8 m^{2}}{8} = m^{2}, \frac{-40 m}{8} = -5m, \frac{16}{8} = 2 \] Thus, the polynomial can be written as: \[ 8(m^{2} - 5m + 2) \]

Key Concepts

Greatest Common Factor

The first step in factoring polynomials is identifying the Greatest Common Factor (GCF). This involves finding the highest number that divides evenly into each term of the polynomial. For example, in the polynomial given (8m^2 - 40m + 16), the coefficients are 8, -40, and 16. The GCF of these coefficients is 8.

In general, here are steps to find the GCF:

• List the factors of each coefficient.
• Identify the highest common factor present in all lists.

Furthermore, if variables are involved, include the variable with the smallest power common to each term. This helps in simplifying the polynomial by factoring out the GCF.Once the GCF is identified, we divide each term of the polynomial by this number to factor it out. This process simplifies the polynomial and makes further factorization easier.

Polynomial Coefficients

Polynomial coefficients are the numerical parts of the terms. For the polynomial 8m^2 - 40m + 16, the coefficients are 8, -40, and 16.

Understanding coefficients is crucial for various operations such as addition, subtraction, multiplication, and factoring of polynomials.

To work with coefficients effectively:

• Identify the coefficients of each term.
• Use them to find common factors, roots, or for simplifying the expression.

Here, we begin by examining the coefficients to identify their GCF, which in our example is 8. Dividing each term of the polynomial by this GCF allows for simplification and easier handling of the polynomial.

Factoring Techniques

Factoring techniques are strategies used to simplify polynomials. Factoring can involve several methods, including finding the GCF, grouping, and applying special formulas.

For the given polynomial 8m^2 - 40m + 16, the most straightforward technique involves factoring out the GCF.

Steps to factor using the GCF technique include:

• Identify the GCF of all terms.
• Divide each term by the GCF.
• Rewrite the polynomial in its factored form.

In our example, after identifying 8 as the GCF, we divide each term by 8 to simplify the polynomial to 8(m^2 - 5m + 2). This is a basic yet powerful technique that paves the way for more complex factoring methods like quadratic factorization or complete factorization.