Problem 30 - Algebra 2

Question

Simplify. $$ (a+6)(a+3) $$

Step-by-Step Solution

Answer

The simplified expression is \(a^2 + 9a + 18\).

Step 1: Use the FOIL Method

To simplify the expression \((a+6)(a+3)\), use the FOIL method. FOIL stands for First, Outside, Inside, Last, representing the order in which you multiply the terms in each binomial.

Step 2: Multiply the First Terms

Multiply the first terms of each binomial: \(a\times a = a^2\).

Step 3: Multiply the Outside Terms

Multiply the outside terms of the binomials: \(a\times 3 = 3a\).

Step 4: Multiply the Inside Terms

Multiply the inside terms of the binomials: \(6\times a = 6a\).

Step 5: Multiply the Last Terms

Multiply the last terms of each binomial: \(6\times 3 = 18\).

Step 6: Add All the Results

Combine the results from the FOIL method: \(a^2 + 3a + 6a + 18\).

Step 7: Combine Like Terms

Combine the like terms \(3a\) and \(6a\) to get \(9a\). So, the simplified expression is \(a^2 + 9a + 18\).

Key Concepts

binomial multiplication

The term "binomial" refers to a polynomial with two terms, typically connected by either addition or subtraction. A common task in algebra is multiplying two binomials, which requires combining each term of the first binomial with each term of the second.

The FOIL method is a straightforward way to handle this multiplication. The acronym FOIL stands for:

• First: Multiply the first terms from each binomial.
• Outside: Multiply the terms on the outer edges of the binomials.
• Inside: Multiply the internal terms of the binomials.
• Last: Multiply the last terms from each binomial.

By applying FOIL, you ensure that every term in both binomials is considered, resulting in a complete product that includes all necessary components. This systematic approach simplifies the process and reduces errors. Always take care to maintain order and keep track of positive and negative signs!

polynomials

In mathematics, a polynomial is an expression consisting of variables, coefficients, and exponents, connected by addition, subtraction, and multiplication. Each part separated by a '+' or '-' sign is known as a term. When multiplied, binomials form polynomials.

In this exercise, multiplying the binomials \((a+6)(a+3)\) results in the polynomial \(a^2 + 9a + 18\).

This polynomial has three terms:

• \(a^2\) - the product of the first terms (First in FOIL).
• \(9a\) - arising from combining the outside and inside products (Outside and Inside in FOIL).
• \(18\) - the result of multiplying the last terms (Last in FOIL).

Polynomials are foundational in algebra because they represent a broad category of expressions. They allow you to explore concepts like factoring, graphing, and solving equations further. Mastering operations with polynomials, including multiplication, is crucial for deeper mathematical understanding.

simplification steps

After multiplying binomials to form a polynomial, the next essential task is to simplify the expression. Simplification is the process of combining like terms and reducing expressions to their simplest form for easier comprehension and further use.

In our expression \(a^2 + 3a + 6a + 18\), simplification steps include:

• **Combining Like Terms**: Here, the like terms are \(3a\) and \(6a\), which simplify to \(9a\).
• **Rewriting the Polynomial**: The expression then becomes \(a^2 + 9a + 18\).

This step is crucial because it ensures the polynomial is concise, allowing for easier interpretation and further analysis or manipulation. Always double-check for possible simplifications by ensuring all like terms are combined and all mathematical operations are executed accurately.