Multiplying two binomials means multiplying each term in the first binomial by each term in the second binomial. FOIL is a memory aid that helps you organize the four products: First, Outer, Inner, and Last. It is useful because it helps prevent missing terms, especially the middle terms that often combine.
For example, (x + 3)(x - 2) becomes x^2 - 2x + 3x - 6, which simplifies to x^2 + x - 6.
The idea behind FOIL is the distributive property used twice. Each term in one set of parentheses must be paired with each term in the other set of parentheses. The Outer and Inner products often create like terms, such as -2x and 3x, which combine into x.
FOIL works only for multiplying two binomials, but the bigger idea of distributing every term works for all polynomial multiplication.
Understanding Math: How to multiply two binomials (FOIL)
One useful way to understand binomial multiplication is to picture a rectangle. Suppose one side has a length of x plus 3 units and the other has a length of x minus 2 units. Split each side at the plus or minus part.
The large rectangle becomes four smaller rectangles. Each small area comes from one length on the first side times one length on the second side. Adding the four areas gives the total area.
This picture explains why every possible pair must be included. A missing product is like leaving out one piece of the rectangle, so the total area would be wrong.
Signs deserve careful attention because they control the middle and constant terms. A positive times a positive is positive. A positive times a negative is negative.
A negative times a positive is negative. A negative times a negative is positive. It helps to decide the sign before multiplying the numbers or variables.
Exponents follow their own rule. When the same variable is multiplied by itself, its exponents add. So x times x becomes x squared.
A number times x stays a term with x. Students sometimes turn x times x into two x, but two x means x plus x, not x multiplied by x.
The result of multiplying two binomials with x as the main variable usually has three parts. There is an x squared term, an x term, and a constant term with no variable. The number in front of x squared comes from the variable terms.
The final constant comes from multiplying the number terms. The middle coefficient is especially important because it is built from two separate cross-products that may cancel, add, or leave a negative result.
This structure matters later in factoring. Factoring reverses the process by finding two binomials that multiply to make a given quadratic expression.
This skill appears whenever a quantity has two changing parts. A rectangle may gain a border, a garden may have its length and width adjusted, or an algebra model may describe area using variable side lengths. In school, binomial products lead into quadratic equations, graphing parabolas, and simplifying formulas in science.
Check your work in a steady order. Write all four products before combining anything. Keep negative signs attached to their terms.
Combine only like terms, meaning terms with exactly the same variable part. A quick final check is to substitute a simple number for x in both the original product and the simplified result. Matching values show that the expansion is likely correct.
Key Facts
- FOIL means First, Outer, Inner, Last.
- (a + b)(c + d) = ac + ad + bc + bd.
- For (x + 3)(x - 2), First: x · x = x^2.
- For (x + 3)(x - 2), Outer and Inner: x · (-2) = -2x and 3 · x = 3x.
- For (x + 3)(x - 2), Last: 3 · (-2) = -6.
- x^2 - 2x + 3x - 6 = x^2 + x - 6.
Vocabulary
- Binomial
- A binomial is an algebraic expression with exactly two terms, such as x + 3 or x - 2.
- FOIL
- FOIL is a mnemonic for First, Outer, Inner, and Last, the four products used when multiplying two binomials.
- Distributive property
- The distributive property says that multiplying a sum means multiplying each term in the sum, such as a(b + c) = ab + ac.
- Like terms
- Like terms have the same variable part and exponent, so they can be combined by adding or subtracting coefficients.
- Coefficient
- A coefficient is the number multiplying a variable, such as 3 in 3x.
Common Mistakes to Avoid
- Multiplying only First and Last is wrong because it skips the Outer and Inner products. For (x + 3)(x - 2), that would miss -2x and 3x.
- Dropping the negative sign in x · (-2) is wrong because the sign is part of the term. The correct product is -2x, not 2x.
- Forgetting to combine like terms is wrong because the expanded form may not be simplified. In x^2 - 2x + 3x - 6, the terms -2x and 3x combine to x.
- Using FOIL on expressions that are not two binomials is wrong because FOIL names only four products. For larger polynomials, use the distributive property to multiply every term by every term.
Practice Questions
- 1 Multiply and simplify: (x + 5)(x + 2).
- 2 Multiply and simplify: (2x - 3)(x + 4).
- 3 Explain why (x + 3)(x - 2) is not equal to x^2 - 6, and identify the missing products.