Problem 57 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (a+5)(a+4) $$

Step-by-Step Solution

Answer

The result of \((a+5)(a+4)\) is \(a^2 + 9a + 20\).

Step 1: Identify the Type of Expression

The given expression is \((a+5)(a+4)\). This is a multiplication of two binomials, where each binomial contains two terms. To solve this multiplication, you should use the distributive property, commonly known as the FOIL method.

Step 2: Apply the FOIL Method

FOIL stands for First, Outer, Inner, Last, which refers to the order of multiplying the terms in the binomials:- **First:** Multiply the first terms in each binomial: \(a \times a = a^2\).- **Outer:** Multiply the outer terms: \(a \times 4 = 4a\).- **Inner:** Multiply the inner terms: \(5 \times a = 5a\).- **Last:** Multiply the last terms: \(5 \times 4 = 20\).

Step 3: Combine Like Terms

Combine the results from the FOIL method:- The expression becomes: \(a^2 + 4a + 5a + 20\).- Combine the like terms \(4a + 5a\): \4a + 5a = 9a.\- The final expression is: \a^2 + 9a + 20.\.

Key Concepts

FOIL Method

The FOIL method is a technique used to multiply two binomials. The acronym FOIL stands for First, Outer, Inner, and Last. It guides you through the sequential process of multiplication when dealing with two-term expressions.
To apply the FOIL method, you:

• Multiply the first terms of each binomial together.
• Then, multiply the outer terms.
• Next, multiply the inner terms.
• Finally, multiply the last terms of each binomial.

By following these steps, you ensure that each term is correctly handled in the multiplication process, providing a structured approach to multiplying binomials.

Binomials

Binomials are algebraic expressions that consist of exactly two terms. For example, in the expression \(a+5\), you have two parts: the variable term \(a\) and the constant term \(5\).
Each part of a binomial can be different, either containing coefficients, variables, or just plain numbers.
When solving problems involving binomials, it’s essential to understand how each term interacts with others, especially during operations like addition, subtraction, or multiplication.
This foundational understanding is crucial for effectively applying strategies such as the FOIL method.

Distributive Property

The distributive property is an essential property of numbers and algebra that allows for the multiplication of a single term by two or more terms inside a parenthesis. It is expressed as \(a(b+c) = ab + ac\).
When multiplying binomials, this property becomes vital as it governs the multiplication process, ensuring that each term in the first binomial multiplies with each term in the second binomial.
The FOIL method is a practical application of this property, breaking down complex multiplications into manageable steps.
Embracing the distributive property helps build a solid understanding of how numbers interact, making it easier to simplify complex algebraic expressions.

Combining Like Terms

Combining like terms is a fundamental step in simplifying algebraic expressions. It involves merging terms that have the same variable raised to the same power. For example, in the expression \(4a + 5a + 20\), the terms \(4a\) and \(5a\) are like terms because they have the same variable \(a\).
Adding them together simplifies the expression to \(9a\), making it neater and more concise.
This process allows for reducing complex equations and is crucial in reaching a simplified final result when using techniques such as the FOIL method.
Understanding how to accurately identify and combine like terms effectively streamlines problem-solving in algebra.