Problem 28 - Elementary Algebra

Question

Factor the greatest common factor from each polynomial. \(45 b-18\)

Step-by-Step Solution

Answer

The polynomial factors to \[9(5b-2)\].

Step 1: Identify the greatest common factor (GCF)

Find the greatest common factor of the terms in the polynomial. For the polynomial \(45b - 18\), determine the GCF of 45 and 18 by finding the largest number that divides both coefficients.

Step 2: Determine the GCF

The factors of 45 are 1, 3, 5, 9, 15, and 45. The factors of 18 are 1, 2, 3, 6, 9, and 18. The greatest common factor of 45 and 18 is 9.

Step 3: Factor out the GCF

Divide each term in the polynomial by the GCF (which is 9) and factor it out: \[45b - 18 = 9(5b - 2)\]

Key Concepts

Greatest Common Factor (GCF)

The concept of the Greatest Common Factor (GCF) is fundamental in algebra. It helps simplify polynomials, making them easier to work with. The GCF is the largest number that divides two or more numbers without leaving a remainder.

To find the GCF of two numbers, let's say 45 and 18:

• List the factors of each number.
• Identify the common factors.
• Select the greatest among the common factors.

For 45, the factors are 1, 3, 5, 9, 15, and 45. For 18, the factors are 1, 2, 3, 6, 9, and 18. The common factors are 1, 3, and 9, hence the greatest one is 9.

This process is crucial before factoring any polynomial as it allows you to simplify the expression and find common terms shared by the entire polynomial.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, connected by addition, subtraction, and multiplication. A polynomial can have multiple terms, and each term is composed of a coefficient and a variable raised to a non-negative integer power.
For example, in the polynomial expression \(45b - 18\), the terms are \45b\ and \18\. Here, 45 and -18 are coefficients, and \b\ is the variable.

Understanding polynomials involves:

• Identifying individual terms
• Recognizing the coefficients and variables
• Combining like terms (terms with the same variables and exponents)

This knowledge is essential when factoring polynomials because you need to be able to decompose a polynomial into simpler terms using the GCF or other factoring techniques.

Factoring

Factoring is the process of breaking down a polynomial into simpler components, which, when multiplied together, give back the original polynomial. This method is useful in algebra for simplifying polynomials, solving polynomial equations, or further analyzing the expressions.

Let's apply factoring to the exercise polynomial \(45b - 18\):

• First, identify the GCF of the polynomial's coefficients (45 and 18). From our previous section, we know the GCF is 9.
• Next, divide each term by the GCF:

For \45b \, dividing by 9, we get \5b \.
For \18 \, dividing by 9, we get \2 \.
So, \[45b - 18 = 9(5b - 2)\] .

Factoring out the GCF simplifies the polynomial and can make further algebraic operations much easier.