Problem 21 - College Algebra

Question

For the following exercises, find the product. $$ (9 v-11)(11 v-9) $$

Step-by-Step Solution

Answer

The product is \(99v^2 - 202v + 99\).

Step 1: Apply the FOIL Method

To find the product of two binomials, use the FOIL (First, Outer, Inner, Last) method. This involves multiplying each term in the first binomial by each term in the second binomial. So, we have: First: Multiply the first terms of each binomial: \(9v \times 11v = 99v^2\).Outer: Multiply the outer terms: \(9v \times -9 = -81v\).Inner: Multiply the inner terms: \(-11 \times 11v = -121v\).Last: Multiply the last terms: \(-11 \times -9 = 99\).

Step 2: Combine Like Terms

Now, combine all the terms obtained from the FOIL method: \(99v^2 - 81v - 121v + 99\).Combine the like terms (the terms with \(v\)): \(-81v - 121v = -202v\).Thus, the expression simplifies to:\(99v^2 - 202v + 99\).

Key Concepts

Understanding Binomials

In algebra, a **binomial** is a polynomial with exactly two terms. Understanding binomials is crucial for operations like addition, subtraction, and especially multiplication using methods like FOIL. Binomials take the form \( ax + b \) where \( a \) and \( b \) are constants or variables.
Consider the expression \((9v - 11)\). It consists of two terms: \(9v\) and \(-11\). Recognizing these separate parts helps in applying multiplication methods such as the FOIL method efficiently.

• Each binomial expression consists of two terms.
• Recognizing these terms is essential for accurate multiplication of binomials.

The concept of binomials extends to operations, where knowing each part makes it easier for performing tasks like expanding expressions.

Like Terms in Polynomial Expressions

When combining results after using the FOIL method or any polynomial expansion, identifying **like terms** is a key step. Like terms in polynomials have the same variables raised to the same power. They can be added or subtracted.
In the expression obtained from our exercise, \(99v^2 - 81v - 121v + 99\), terms with \(v^2\), terms with \(v\), and constant terms need grouping.
For instance, the terms \(-81v\) and \(-121v\) are like terms because both have the variable \(v\) to the power of one.

• The power and variable must match for terms to be "like".
• Combining like terms simplifies expressions.
• This simplification involves adding or subtracting coefficients.

Identifying and combining like terms is an essential skill for simplifying polynomial expressions and keeping them manageable.

The FOIL Method and Polynomial Multiplication

One essential technique for multiplying binomials is the **FOIL Method**. FOIL stands for First, Outer, Inner, Last, referring to the order of multiplying terms.
Let's take an example: multiplying \((9v - 11)(11v - 9)\). By applying the FOIL method:

• **First:** Multiply the first terms of each binomial: \(9v \times 11v = 99v^2\).
• **Outer:** Multiply the outer terms: \(9v \times -9 = -81v\).
• **Inner:** Multiply the inner terms: \(-11 \times 11v = -121v\).
• **Last:** Multiply the last terms: \(-11 \times -9 = 99\).

The result of these multiplications is then combined by adding or subtracting them, based on their signs, to yield a single polynomial.
Polynomial multiplication, especially using the FOIL method, simplifies the expansion of binomials into longer expressions. This expansion becomes crucial in algebraic problem-solving where multiplying two-term expressions is necessary.