Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises, factor the polynomial. If the polynomial is prime, state it. $$ 3 m^{3}+3 m^{2}-18 m $$

Step-by-Step Solution

Answer

The factored form of the given polynomial is: \(3m(m + 3)(m - 2)\).

Step 1: : The given polynomial is \(3m^3 + 3m^2 - 18m\) To find the GCF, let's find the common factors for each term: 3m^3: \(1, 3, m, 3m, m^2, 3m^2, m^3, 3m^3\) 3m^2: \(1, 3, m, 3m, m^2, 3m^2\) -18m: \(1, 2, 3, 6, 9, 18, m, 2m, 3m, 6m, 9m, 18m\) The GCF is 3m. #step_2# - Factoring out the GCF

: Now that we have identified the GCF (3m), we can factor it out from the given polynomial: \(3m (m^2 + m - 6)\) #step_3# - Factoring the remaining expression

Step 2: : The remaining expression inside the parenthesis is a quadratic expression: \(m^2 + m - 6\) In order to factor it, we need to find two numbers whose product is equal to the constant term (-6) and whose sum is equal to the coefficient of the linear term (1). These two numbers are 3 and -2, since: \(3 \times -2 = -6\) and \(3 + (-2) = 1\) So, we can factor the expression as follows: \(m^2 + m - 6 = (m + 3)(m - 2)\) #step_4# - Final factored form

: Putting everything together, we have factored the given polynomial as: \(3m^3 + 3m^2 - 18m = 3m(m + 3)(m - 2)\) So the factored form of the given polynomial is: \(3m(m + 3)(m - 2)\)

Key Concepts

GCF (Greatest Common Factor)

The first step in factoring polynomials involves finding the Greatest Common Factor (GCF). This is the largest factor shared by all terms in the polynomial. In our exercise, the polynomial is given by \(3m^3 + 3m^2 - 18m\). To identify the GCF, we need to examine each term:

• \(3m^3\) has factors of 3 and \(m^3\).
• \(3m^2\) shares factors with \(3m^3\), namely 3 and \(m^2\).
• \(-18m\) includes factors such as 3 and \(m\) as well.

All terms include 3 and \(m\), making \(3m\) the GCF.
Once found, we can "factor" the GCF out of each term. This process simplifies the polynomial and lays the foundation for more detailed factoring.

Quadratic Expression

After extracting the GCF, the polynomial reduces to a quadratic expression in parentheses: \(m^2 + m - 6\).
A quadratic expression is one that involves a variable squared, like \(m^2\). It typically takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
In this case, \(a = 1\), \(b = 1\), and \(c = -6\). Quadratics are essential because they commonly appear in various mathematical contexts, such as motion and area problems. A key step in solving these expressions is factoring, which allows us to express them as a product of simpler expressions.

Factored Form

To convert a quadratic expression into its factored form, look for two numbers that:

• Multiply to the constant term, \(c\) (in our example, \(-6\)).
• Add up to the linear coefficient, \(b\) (here, \(1\)).

For \(m^2 + m - 6\), these numbers are 3 and -2.
Thus, the quadratic \(m^2 + m - 6\) can be rewritten as \((m + 3)(m - 2)\).
By factoring, we've expressed the quadratic as a product of two binomials, resulting in a factored form that's more manageable—for both solving and understanding the polynomial's behavior.

Polynomial Expression

A polynomial expression contains terms that are combinations of variables raised to whole number powers, each scaled by coefficients. Polynomials can be as simple as \(x^2\) or as complex as \(3m^3 + 3m^2 - 18m\).
To work with polynomials effectively, factorization is a key skill. It helps break down a complex polynomial into simpler components, revealing roots and simplifying equations further down the line.
In the given problem, after identifying the GCF and factoring the quadratic, the polynomial is transformed into \(3m(m + 3)(m - 2)\).
Such transformations make it easier to solve equations or evaluate polynomial functions for particular values, vital in mathematics and its applications.