Multiplying Binomials and Special Products
Multiplying two binomials always produces four pieces before anything combines, and squaring a binomial always leaves a middle term, since (a+b)2 means (a+b) times (a+b), not a2 plus b2.
When you multiply (x + 3)(x + 5), every term in the first binomial has to meet every term in the second. That's where FOIL comes from: First times First, Outer times Outer, Inner times Inner, Last times Last. Four multiplications, always, because a sum times a sum means each piece of one sum gets distributed across each piece of the other. Picture a rectangle with sides x + 3 and x + 5. Its area splits into four smaller rectangles: x times x, x times 5, 3 times x, and 3 times 5. Add those four tiles and you get x2 + 8x + 15 once the two middle tiles combine.
Try (2x - 3)(x + 7). First: 2x times x = 2x2. Outer: 2x times 7 = 14x. Inner: -3 times x = -3x. Last: -3 times 7 = -21. The middle terms 14x and -3x combine to 11x, giving 2x2 + 11x - 21. That combining step is where most of the work happens once the four pieces are laid out correctly.
Squaring a binomial is just FOIL in disguise, since (a + b)2 means (a + b)(a + b). FOIL that out and you get a2 + ab + ba + b2, which is a2 + 2ab + b2 once the two middle terms combine. Skipping straight to a2 + b2 throws away that middle term entirely. This perfect-square pattern, along with the difference-of-squares pattern (a+b)(a-b) = a2 - b2, shows up constantly when factoring quadratics or simplifying rational and radical expressions elsewhere in algebra.
Key terms
- FOIL
- A method for multiplying two binomials by combining First, Outer, Inner, and Last term products.
- Binomial
- An expression with exactly two terms added or subtracted, such as x + 3.
- Perfect-square pattern
- The rule that (a + b)2 equals a2 + 2ab + b2, skipping full FOIL.
- Difference of squares
- The pattern (a + b)(a - b) = a2 - b2, where the middle terms cancel instead of combining.
- Like terms
- Terms with the same variable raised to the same power, which can be added or subtracted together.
Watch it explained
- How it worksHow does a rectangle with sides x + 3 and x + 5 lay FOIL out as four tiles of area?
- Worked exampleFind the product of (2x - 3) and (x + 7) and combine the two middle terms.
- Common mistakeWhy isn't (a + b) squared equal to a squared plus b squared when squaring hits the whole thing?
- Worked exampleFind (3x + 4) squared using the perfect-square pattern instead of multiplying it out.
Check yourself
End of topic.