Problem 40 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using the FOIL method, and simplify. \((s+8)(s-2)\)

Step-by-Step Solution

Answer

The product is \(s^2 + 6s - 16\).

Step 1: Apply FOIL Method - Multiply First Terms

The first step in the FOIL method is to multiply the first terms of each binomial. In our expression \((s+8)(s-2)\), the first terms are \(s\) and \(s\). So, we multiply them: \(s \times s = s^2\).

Step 2: Apply FOIL Method - Multiply Outer Terms

Next, multiply the outer terms of the binomials. In our expression, the outer terms are \(s\) and \(-2\). Thus, we calculate: \(s \times (-2) = -2s\).

Step 3: Apply FOIL Method - Multiply Inner Terms

Now, multiply the inner terms from each binomial. For \((s+8)(s-2)\), the inner terms are \(8\) and \(s\). Therefore, we compute: \(8 \times s = 8s\).

Step 4: Apply FOIL Method - Multiply Last Terms

Finally, multiply the last terms of each binomial. The last terms here are \(8\) and \(-2\). So, the multiplication is: \(8 \times (-2) = -16\).

Step 5: Combine Like Terms

Now, combine all the results from the FOIL steps: \(s^2\) from the first terms, \(-2s\) from the outer terms, \(8s\) from the inner terms, and \(-16\) from the last terms. The expression is: \(s^2 - 2s + 8s - 16\).

Step 6: Simplify the Expression

Combine the like terms in the expression. Combine \(-2s\) and \(8s\) to get \(6s\). The simplified expression is: \(s^2 + 6s - 16\).

Key Concepts

Understanding Binomial Multiplication

Binomial multiplication is a fundamental concept in algebra, dealing with the multiplication of polynomials that have exactly two terms, known as binomials. A common example is the binomial

• Expression: \((a + b)(c + d)\)
• Each binomial consists of two terms.

The goal of multiplying binomials is to expand them into a single polynomial. For this purpose, we frequently use the FOIL method, which stands for First, Outer, Inner, Last and specifies the order of multiplying the terms.
Using the FOIL method helps ensure that each term from the first binomial is multiplied by each term from the second binomial in a systematic way, thereby avoiding mistakes. It's important to note that when multiplying, you'll often create four products which then you need to combine and simplify.

Exploring Algebraic Expressions

An algebraic expression is composed of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. Variable symbols (like \(x\), \(y\), or \(s\)) are used to represent numbers whose values are not immediately known. These expressions are vital in forming equations and formulas in math and can vary in complexity.
Algebraic expressions can be as simple as a single term, called a "monomial," such as \(3x\), or they can include polynomial expressions with multiple terms, like a "binomial" or a "trinomial."
In the context of the problem

• Expression: \((s + 8)(s - 2)\)
• The terms: \(s\) and numbers - these represent known and variable components.

Algebraic expressions are manipulated by applying rules of arithmetic and algebra to solve for values or to simplify the expressions into a basic form.

Simplification of Expressions

Simplifying expressions involves reducing them into their simplest form, where no further arithmetic can be performed. After performing binomial multiplication using the FOIL method, you get several terms. Combining these and simplifying ensures the expression is presented in the clearest and most compact form possible.
In the expression

• Combine like terms: Terms with the same variable and exponent.

After performing all multiplications in the step-by-step solution, we obtained:

• Result: \(s^2 - 2s + 8s - 16\)
• Combine: \(-2s + 8s = 6s\)
• Final simplified expression: \(s^2 + 6s - 16\)

Simplification helps in understanding the expression's behavior and in solving equations or evaluating the expression for specific variable values.