Problem 73 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ (a+4)(a+2) $$

Step-by-Step Solution

Answer

Question: Multiply the binomials (a+4)(a+2) and simplify the result. Answer: a^2+6a+8

Step 1: Identify the binomials

We are given two binomials \((a+4)(a+2)\). We will use the distributive property to multiply them together.

Step 2: Apply the distributive property (FOIL method)

The distributive property states that for any numbers a, b, and c: \((a+b)c = ac + bc\). The FOIL method is simply the distributive property applied twice. Here, it stands for First, Outer, Inner, Last. First: Multiply the first terms of both binomials together: \((a)(a)=a^2\) Outer: Multiply the outer terms of both binomials together: \((a)(2)=2a\) Inner: Multiply the inner terms of both binomials together: \((4)(a)=4a\) Last: Multiply the last terms of both binomials together: \((4)(2)=8\) Now, we can write the sum of these products: \(a^2+2a+4a+8\)

Step 3: Combine like terms

Now, combine the like terms (the terms with the same power of 'a'): \(a^2+(2a+4a)+8\) Simplify the expression by combining the coefficients for the linear terms: \(a^2+6a+8\)

Step 4: State the final result

The multiplication of the given binomials \((a+4)(a+2)\) is equal to the simplified expression \(a^2+6a+8\).

Key Concepts

FOIL Method

When faced with multiplying two binomials like \((a+4)(a+2)\), the FOIL method is a handy tool. FOIL stands for:

• First: Multiply the first terms of each binomial. For \((a+4)(a+2)\), this would be \(a \cdot a\), giving \(a^2\).
• Outer: Multiply the outer terms of the expression. Here, that’s \(a \cdot 2\), resulting in \(2a\).
• Inner: Multiply the inner terms. In this case, \(4 \cdot a\), giving you \(4a\).
• Last: Multiply the last terms of each binomial. This is \(4 \cdot 2\), equaling \(8\).

By writing down each product, you get \(a^2 + 2a + 4a + 8\). It’s simply the distributive property in action for each part of the binomial.

Distributive Property

The distributive property is a guiding principle in algebra. It helps you multiply a single term and two or more terms inside a bracket. Here’s how it works:- Imagine you have \((a+b)c\). Using the distributive property, this becomes \(ac + bc\).- With two binomials like \((a+4)(a+2)\), the property is applied twice: - First, distribute \((a+4)\) across \(a\) to get \(a^2 + 4a\). - Next, distribute the \((a+4)\) across \(2\) to get \(2a + 8\).
Combining everything will bring us back to the expression \(a^2 + 2a + 4a + 8\). This shows how each part contributes to the final equation.

Combining Like Terms

Once you've multiplied the binomials using the FOIL method or distributive property, the next step is to combine like terms. Like terms have the same variable raised to the same power.- In the expression \(a^2 + 2a + 4a + 8\), identify and group the terms with the same variable: - The terms \(2a\) and \(4a\) are like terms because they both have the variable \('a'\) to the first power. - Combine them by adding their coefficients: \(2a + 4a = 6a\).
Put it all together, and you'll have the simplified expression \(a^2 + 6a + 8\). This process tidies up your equation, making it easier to understand and use.