Problem 29 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ (2 x+3)(3 x-2) $$

Step-by-Step Solution

Answer

The simplified expression is: \(6x^2 + 5x - 6\).

Step 1: Identify the terms

The given expression consists of two terms: \((2x + 3)\) and \((3x - 2)\).

Step 2: Apply the distributive property (FOIL method)

FOIL stands for First, Outer, Inner, Last. We will multiply the terms using this method: First: Multiply the first terms of each term: \((2x)(3x)\) Outer: Multiply the outer terms of each term: \((2x)(-2)\) Inner: Multiply the inner terms of each term: \((3)(3x)\) Last: Multiply the last terms of each term: \((3)(-2)\)

Step 3: Calculate each product

Now, we will calculate each of the products from Step 2: First: \((2x)(3x) = 6x^2\) Outer: \((2x)(-2) = -4x\) Inner: \((3)(3x) = 9x\) Last: \((3)(-2) = -6\)

Step 4: Combine the products and simplify

Combine the products found in Step 3 and simplify by combining like terms: \(6x^2 - 4x + 9x - 6\) Combine like terms (-4x and 9x): \(6x^2 + 5x - 6\) The resulting simplified expression is: \(6x^2 + 5x - 6\)

Key Concepts

Distributive Property

When multiplying polynomials, one of the key concepts involved is the distributive property. This property allows us to multiply each term in one polynomial by each term in another polynomial. It's fundamental in handling expressions like \((a + b)(c + d)\), because we handle each part of the expressions individually to ensure nothing is forgotten.

To break it down: if we have two binomials like \((2x + 3)\) and \((3x - 2)\), using the distributive property, we multiply:

• Each term in the first binomial by each term in the second binomial.
• This includes setting up the products in sequences found in methods like FOIL (which we'll cover next).

This approach ensures you account for every combination of terms, leading us gradually to a simplified expression.

FOIL Method

The FOIL method is a handy tool for multiplying two binomials. It's a mnemonic that stands for: First, Outer, Inner, Last. This method ensures that you combine all parts of the two expressions correctly.

Let's apply it to our example, \((2x + 3)(3x - 2)\):

• First: Multiply the first terms: \(2x \times 3x = 6x^2\)
• Outer: Multiply the outer terms: \(2x \times -2 = -4x\)
• Inner: Multiply the inner terms: \(3 \times 3x = 9x\)
• Last: Multiply the last terms: \(3 \times -2 = -6\)

By systematically following this method, we ensure that no term combinations are overlooked. It simplifies the process of polynomial multiplication, making it intuitive and methodical.

Combining Like Terms

After multiplying the terms using the distributive property or FOIL method, the next step is to simplify the expression by combining like terms.

In our expression \(6x^2 - 4x + 9x - 6\), there are some terms we can combine:

• The terms \(-4x\) and \(9x\) are 'like terms' because they both contain the variable \(x\) raised to the first power. When combined, they give \(5x\).

Thus, the simplified expression becomes:

• \(6x^2 + 5x - 6\)

This step is crucial as it reduces the complexity of the polynomial and provides the final simplified form, making it easier to analyze or use in further calculations.