Problem 56 - Beginning and Intermediate Algebra

Question

Use FOIL to multiply. $$ (c+5)(8-c) $$

Step-by-Step Solution

Answer

The simplified expression after using the FOIL method to multiply \((c+5)(8-c)\) is \(-c^2 + 3c + 40\).

Step 1: First Terms

Multiply the first terms of each binomial, which are c and 8. This gives us: \(c \cdot 8 = 8c\)

Step 2: Outer Terms

Multiply the outer terms of each binomial, which are c and -c. This gives us: \(c \cdot (-c) = -c^2\)

Step 3: Inner Terms

Multiply the inner terms of each binomial, which are 5 and 8. This gives us: \(5 \cdot 8 = 40\)

Step 4: Last Terms

Multiply the last terms of each binomial, which are 5 and -c. This gives us: \(5 \cdot (-c) = -5c\)

Step 5: Combine Results

Add all the results from steps 1 to 4 to get the final expression: \(8c - c^2 + 40 - 5c\)

Step 6: Simplify the Expression

Combine the like terms (8c and -5c) to get the simplified expression: \(-c^2 + 3c + 40\) So, the final result after using the FOIL method to multiply (c+5)(8-c) is: \((c+5)(8-c) = -c^2 + 3c + 40\)

Key Concepts

Understanding Binomials

A binomial is simply an algebraic expression containing two terms separated by a plus or minus sign. In the expression \((c+5)(8-c)\), each section inside the parentheses is a binomial. The key idea here is that each part consists of:

• One term (either a constant or a variable) added or subtracted to another term.
• Binomials can represent a wide range of mathematical scenarios, from simple numeric expressions to complex models in calculus.

Understanding binomials is essential in algebra as they often serve as building blocks for more complex expressions. Learning how to manipulate them with operations like addition, subtraction, and especially multiplication will greatly enhance your problem-solving skills.

Exploring Polynomial Multiplication with FOIL

Polynomial multiplication involves multiplying variables or terms in one polynomial expression by every term in another. The FOIL method is a systematic way to multiply two binomials, standing for First, Outer, Inner, Last—referring to which terms to multiply.

When you apply FOIL to \((c+5)(8-c)\), you're essentially following these steps:

• First: Multiply the first terms of each binomial: \(c \cdot 8 = 8c\).
• Outer: Multiply the outermost terms: \(c \cdot (-c) = -c^2\).
• Inner: Multiply the inner terms: \(5 \cdot 8 = 40\).
• Last: Multiply the last terms: \(5 \cdot (-c) = -5c\).

This structured approach easily organizes your work and reduces errors. Once each multiplication is done, you combine like terms to simplify the result. Recognizing patterns and correctly forming products can make even complex expressions manageable.

Navigating Algebraic Expressions

An algebraic expression is a combination of numbers and letters called variables, elevated by operations like addition, subtraction, multiplication, and division.
In our example, \((c+5)(8-c)\), you're dealing with an algebraic expression that's expanded to form \(-c^2 + 3c + 40\).

Key parts to understand:

• Variables: These are symbols that represent unknown values—like \(c\) in this exercise.
• Coefficients: Numbers placed before variables to indicate multiplication, such as \(8\) in \(8c\).
• Constants: Fixed numbers that remain the same, like the \(40\) in the result.

Grasping algebraic expressions allows you to predict outcomes and simplify problems. Understanding how to read, dissect, and reassemble them is fundamental in solving equations and application in real-life scenarios.