Problem 42 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using the FOIL method, and simplify. \((4 s-1)(2 s+5)\)

Step-by-Step Solution

Answer

The simplified expression is \(8s^2 + 18s - 5\).

Step 1: Apply the FOIL Method

The FOIL method stands for First, Outer, Inner, Last. Multiply the first terms of each binomial: \[(4s) \times (2s) = 8s^2\]

Step 2: Multiply the Outer Terms

Next, multiply the outer terms of the binomials:\[ (4s) \times 5 = 20s \]

Step 3: Multiply the Inner Terms

Now multiply the inner terms of the binomials:\[ (-1) \times (2s) = -2s \]

Step 4: Multiply the Last Terms

Finally, multiply the last terms in each binomial:\[ (-1) \times 5 = -5 \]

Step 5: Combine the Results

Now add all the results from each of the steps to combine them:\[ 8s^2 + 20s - 2s - 5 \]

Step 6: Simplify the Expression

Combine like terms to simplify the expression:\[ 8s^2 + 18s - 5 \]

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental component in mathematics. They consist of variables, constants, and coefficients connected by mathematical operations like addition, subtraction, multiplication, and division. In our exercise, the expression \((4s-1)(2s+5)\) is an example of an algebraic expression.

Breaking this down, the parts of an expression include:

• Variables: These are symbols that represent numbers, such as \(s\) in the expression.
• Constants: Fixed numbers that do not change, such as \(-1\) and \(5\).
• Coefficients: Numbers multiplying the variables, like \(4\) and \(2\).

Understanding the components of algebraic expressions helps in manipulating and simplifying them through various methods like the FOIL method in this case. Remember, solving algebraic expressions often involves combining like terms and using operations to structure the expression in a simplified form.

Binomials

In algebra, a binomial is an expression containing exactly two terms. The expression \((4s - 1)\) is a binomial, as is \((2s + 5)\). Each of these has two terms separated by either a plus or minus sign.

The purpose of multiplying binomials is to expand and simplify the expression into a new form. This process often uses the FOIL method, which stands for:

• First: Multiply the first terms in each binomial together.
• Outer: Multiply the outer terms in the overall expression.
• Inner: Multiply the inner terms in the expression.
• Last: Multiply the last terms of each binomial.

Using FOIL, you expand \((4s-1)(2s+5)\) step-by-step to get the terms \(8s^2\), \(20s\), \(-2s\), and \(-5\). Having mastered binomials opens many pathways to solving more complex algebraic problems effectively.

Like Terms

Like terms are an important concept when simplifying algebraic expressions. They are terms that contain the exact same variables raised to the same powers, though their coefficients can vary.

In our exercise, once you've applied the FOIL method to find \(8s^2 + 20s - 2s - 5\), the next step is to simplify. Here, you combine the like terms \(20s\) and \(-2s\), resulting in \(18s\).

• It simplifies the expression by reducing the number of terms, making calculations easier.
• It helps in accurately solving and evaluating algebraic equations or expressions.

Remember, identifying and combining like terms involves ensuring that the variables and their exponents match exactly. This skill is crucial for achieving a simplified form, such as \(8s^2 + 18s - 5\), ensuring clearer and more manageable expressions.