Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ (3 r+2 s)(4 r-3 s) $$

Step-by-Step Solution

Answer

The simplified expression is: \( 12r^2 - rs - 6s^2 \)

Step 1: Apply distributive property (FOIL method)

FOIL stands for "First, Outer, Inner, Last," referring to how we multiply the binomial expressions. Expand the expression using the FOIL method: $$ (3r + 2s)(4r - 3s) $$ First, multiply the first terms in each binomial: $$ (3r)(4r) = 12r^2 $$ Next, multiply the outer terms: $$ (3r)(-3s) = -9rs $$ Then, multiply the inner terms: $$ (2s)(4r) = 8rs $$ Finally, multiply the last terms in each binomial: $$ (2s)(-3s) = -6s^2 $$

Step 2: Combine like terms

Now, combine the like terms from the results of the FOIL method: $$ 12r^2 - 9rs + 8rs - 6s^2 $$ Combine the terms with 'rs': $$ 12r^2 - 1rs - 6s^2 $$

Step 3: Write the final answer

The simplified expression is: $$ 12r^2 - rs - 6s^2 $$

Key Concepts

Binomial Expansion

The binomial expansion is a process used to expand expressions that involve two terms (binomials). When multiplying two binomials, each term in the first binomial multiplies with each term in the second. This results in a new expression that combines all these products. In the example

• (3r + 2s)(4r - 3s),

we are essentially expanding the binomial expression to simplify and find its equivalent polynomial form.
The binomial expansion frequently involves concepts like the distributive property or specific techniques such as the FOIL method, which aids in simplifying and organizing the multiplication process. This method is particularly useful because it ensures that no terms are left out in the expansion.
Whether you're working with simple or complex binomials, binomial expansion is a foundational concept in algebra, helping to bridge basic operations with more advanced mathematical challenges.

Distributive Property

The distributive property is a fundamental concept in algebra, notable for its versatility in simplifying expressions. It allows the operation of multiplication to be distributed over addition or subtraction within an expression. Formally, the property is expressed as

• a(b + c) = ab + ac.

When applying this property to the expression

• (3r + 2s)(4r - 3s),

the multiplication of each individual term across the binomials occurs. First, each term in the first binomial multiplies every term in the second. This ensures every possible combination of terms is included in the final expression.
The distributive property is vital for simplifying expressions and solving equations, as it breaks down complex operations into manageable steps. This simplification process is key to ensuring clarity and accuracy in mathematics.

FOIL Method

The FOIL method is a handy technique designed to simplify binomial multiplication. It stands for First, Outer, Inner, Last, representing the four multiplications needed to expand the product of two binomials like

• (3r + 2s)(4r - 3s).

Here's a breakdown of each step:

• **First:** Multiply the first terms from each binomial, resulting in \(3r \times 4r = 12r^2\).
• **Outer:** Multiply the outer terms, \(3r \times -3s = -9rs\).
• **Inner:** Multiply the inner terms, \(2s \times 4r = 8rs\).
• **Last:** Finally, multiply the last terms, \(2s \times -3s = -6s^2\).

The beauty of the FOIL method is its structured approach to ensuring that every term is accounted for, enabling a straightforward path to the final expanded polynomial. After applying FOIL, combine like terms such as \(-9rs\) and \(8rs\) to simplify the expression further, which ultimately results in the answer \(12r^2 - rs - 6s^2\). This method is a practical tool in algebra that makes polynomial multiplication less daunting and more organized.