Multiplying Binomials: Complete Step-by-Step FOIL Guide
A binomial is an algebraic polynomial containing exactly two terms. Multiplying two linear binomials is one of the most fundamental operations in intermediate algebra.
The FOIL Mnemonic Explained
The acronym FOIL ensures that every term in the first binomial multiplies every term in the second binomial via the distributive property:
(ax + b)(cx + d) = \underbrace{ac\,x^2}_{\text{First}} + \underbrace{ad\,x}_{\text{Outer}} + \underbrace{bc\,x}_{\text{Inner}} + \underbrace{bd}_{\text{Last}}
- 1F - First terms: Multiply the first term of each binomial: $(ax) \cdot (cx) = ac\,x^2$.
- 2O - Outer terms: Multiply the outermost terms: $(ax) \cdot (d) = ad\,x$.
- 3I - Inner terms: Multiply the innermost terms: $(b) \cdot (cx) = bc\,x$.
- 4L - Last terms: Multiply the constant ending terms: $(b) \cdot (d) = bd$.
Finally, combine the like linear terms: $ad\,x + bc\,x = (ad + bc)x$.
Visual Geometric Multiplication (Box Method)
Multiplying binomials corresponds geometrically to computing the area of a rectangle of side lengths $(ax + b)$ and $(cx + d)$:
| $\times$ | $cx$ | $d$ |
|---|---|---|
| $ax$ | $ac\,x^2$ | $ad\,x$ |
| $b$ | $bc\,x$ | $bd$ |
Summing the four sub-areas yields the total expanded polynomial area.