Multiplication of Two Binomials (FOIL Method)

TL;DR

When you multiply two binomials, you're essentially distributing each term from the first binomial to each term in the second. The FOIL method is a simple way to remember all the necessary pairings. It ensures you multiply every term exactly once.

1. The Mental Model

Think of multiplying two binomials like shaking hands with everyone at a small party. Each person from the first group needs to shake hands with every person from the second group. FOIL helps you remember each unique handshake.

2. The Core Material

Multiplying two binomials means taking an expression like (a + b) and multiplying it by another expression like (c + d). The result will be an expression with four terms, which you'll then simplify if possible.

What's a Binomial?

A binomial is just an algebraic expression with two terms. For example, (x + 3) is a binomial; (2y - 5) is also a binomial. The terms are separated by a plus or minus sign.

The FOIL Method

FOIL is an acronym that stands for:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outermost terms in the entire expression.
• Inner: Multiply the innermost terms in the entire expression.
• Last: Multiply the last terms in each binomial.

Let's look at how it works with (a + b)(c + d):

graph TD
    start("Start with (a + b)(c + d)") --> F["First Terms: a * c"]
    F --> O["Outer Terms: a * d"]
    O --> I["Inner Terms: b * c"]
    I --> L["Last Terms: b * d"]
    L --> result("Result: ac + ad + bc + bd")

After you perform these four multiplications, you'll add all the results together. Often, the "Outer" and "Inner" terms will be "like terms" (meaning they have the same variable raised to the same power), and you'll be able to combine them to simplify the expression further.

Combining Like Terms

Once you've applied FOIL, you'll end up with four terms. Look for any terms that have the same variable part. For example, 3x and 5x are like terms and can be added to 8x. 2x^2 and 7x^2 are also like terms. However, 4x and 6x^2 are not like terms because the variable x has different powers.

3. Worked Example

Let's multiply (x + 3)(x + 5) using the FOIL method.

1. First: Multiply the first terms in each binomial.
   x * x = x^2

2. Outer: Multiply the outermost terms.
   x * 5 = 5x

3. Inner: Multiply the innermost terms.
   3 * x = 3x

4. Last: Multiply the last terms in each binomial.
   3 * 5 = 15

Now, combine all these results:
x^2 + 5x + 3x + 15

Finally, combine the like terms (5x and 3x):
x^2 + 8x + 15

So, (x + 3)(x + 5) = x^2 + 8x + 15.

4. Key Takeaways

• The FOIL method is a mnemonic to remember the four pairs of terms you need to multiply.
• "First," "Outer," "Inner," and "Last" refer to the positions of the terms in the binomials.
• You always multiply each term from the first binomial by each term from the second.
• After multiplying, you'll sum all four products together.
• Always simplify your answer by combining any like terms.

Common Mistakes to Avoid

• Forgetting to multiply all four pairs of terms, especially the inner or outer terms.
• Making sign errors, particularly when there are negative numbers involved.
• Incorrectly combining like terms (e.g., adding x and x^2).
• Thinking that (a + b)^2 is equal to a^2 + b^2 (it's not – you need to use FOIL on (a + b)(a + b)).

5. Now Try It

Multiply (2y - 4)(y + 7) using the FOIL method and simplify your answer as much as possible.

What to do: Apply each step of FOIL (First, Outer, Inner, Last), write down the four products, and then combine any like terms to get your final simplified expression.

What success looks like: Your final answer should be a trinomial (an expression with three terms) with no like terms left to combine, correctly accounting for all the signs.