You've seen equations like (x + 3)(x + 5). But what does that actually mean? And how do you turn it into something simpler?
That's what FOIL is for. It's a pattern — a recipe — for multiplying two binomials without missing any pieces.
"Bi" = two, "nomial" = terms. A binomial is just two things added or subtracted. You already know "bi" from your root words — bicycle, binocular.
When we write (x + 3)(x + 5), we're multiplying two binomials together. Each one has two parts, so there are 2 × 2 = 4 multiplications to do.
That's all FOIL is — a way to remember which 4 multiplications to do.
First: x · x Outer: x · 5 Inner: 3 · x Last: 3 · 5
You multiply every term in the first parenthesis by every term in the second. FOIL just gives you the order so you don't miss one.
Add them up: x² + 5x + 3x + 15
Tap each step to reveal it. The O and I terms (5x + 3x) combine because they're both "x" terms.
(x + 3)(x + 5) is literally the area of a rectangle with sides (x+3) and (x+5). The four parts of FOIL are four smaller rectangles inside it:
The total area = x² + 5x + 3x + 15 = x² + 8x + 15
Slide a and b to see how the result changes. Notice the pattern — the middle number is always a + b and the last number is always a × b.
FOIL works exactly the same way — you just have to watch your signs.
The sign carries with the number. If it's (x − 2), the 2 is negative everywhere it goes.
For (x + a)(x + b), the answer is always:
The middle coefficient is the sum. The last term is the product. Once you see this pattern, you can FOIL simple ones in your head.
Sum: 4+6=10 Product: 4×6=24
That last one is straight from your EOC review.
FOIL expands: (x+3)(x+5) → x² + 8x + 15
Factoring reverses it: x² + 8x + 15 → (x+3)(x+5)
You already know how to solve x² + 8x + 15 = 0 with the quadratic formula. But if you can factor it back into (x+3)(x+5) = 0, the answer jumps out: x = −3 or x = −5.