Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Factor out the greatest common factor. $$ 9 a b^{2}-6 a^{2} b $$

Step-by-Step Solution

Answer

The simplified expression after factoring out the GCF from the given polynomial \(9ab^2 - 6a^2b\) is \(3ab(3b - 2a)\).

Step 1: Observe the polynomial

We have the polynomial \(9ab^2 - 6a^2b\). Our goal is to factor out the greatest common factor from this expression.

Step 2: Find the GCF of the coefficients

The coefficients of the polynomial are 9 and 6. The GCF of 9 and 6 is 3. So, we will factor out 3 from the coefficients.

Step 3: Find the GCF of the variables

We can see two variables in the polynomial, a and b. For variable a, we have powers 1 and 2. The GCF of powers is the lower power, so the GCF for variable a is \(a\). For variable b, we have powers 2 and 1. The GCF for variable b is \(b\).

Step 4: Factor out the GCF from the polynomial

Now, we have the GCF of the coefficients and variables as 3, \(a\), and \(b\). We can factor out these from the polynomial by factoring out the GCF from the coefficients and variables separately. So, we have: \[3ab(3b - 2a)\] The given polynomial is factored out, and the simplified expression is \(3ab(3b - 2a)\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) of a set of terms is the largest factor that is shared by all the terms. It plays a crucial role in simplifying polynomials.
Finding the GCF eases calculations and helps to factor expressions effectively. For polynomials like the one presented, we begin by identifying the GCF of both the coefficients and the variables separately.

To determine the GCF of the coefficients, inspect their numerical values. The coefficients in the exercise are 9 and 6.

• List the factors of each number:
  • 9: 1, 3, 9
  • 6: 1, 2, 3, 6

The common factors are 1 and 3, and the greatest among them is 3. Thus, 3 is the GCF of the coefficients.

Finding the GCF of a variable involves comparing the exponents across the terms. More about this in the next section.

Variable Exponents

Variable exponents tell us how many times a variable is used as a factor in a term. Understanding exponents is essential when factoring expressions, as it allows us to simplify them efficiently.
In the exercise, we focus on two variables, "a" and "b". Since they appear in both terms, finding their GCF will help us factor the polynomial effectively.

To find the GCF of variables, you should consider the powers (exponents) of the variables in the terms.

• For variable 'a':
  In the terms 9ab² and -6a²b, the powers of 'a' are 1 and 2, respectively.
  The GCF of these powers is the lower power, which is 1, meaning the GCF for 'a' is simply "a".
• For variable 'b':
  The powers are 2 and 1, respectively, for the same terms.
  Here, the smallest power is 1, so the GCF for 'b' is "b".

Hence, the combined GCF for the variables is "ab".

Polynomial Coefficients

Polynomial coefficients are the numerical factors attached to the variables in a polynomial expression. They influence the expression's degree and play a significant role in determining its GCF.

Understanding how to handle polynomial coefficients efficiently is crucial when simplifying or factoring polynomials. In the equation from the exercise, 9 and 6 are our coefficients.
To factor these coefficients, we need to determine the GCF, as discussed earlier.

After finding it, you divide each coefficient by this GCF to simplify the polynomial expression:

• The original terms were 9ab² and -6a²b.
• Dividing 9 by 3 gives us 3, and dividing 6 by 3 gives us 2.

This division helps to form the simplified expression: 3ab(3b - 2a). This operation not only makes the polynomial more manageable but also prepares it for further applications, such as solving equations or analyzing functions.