A way of remmembering how to multiply two binomials. Stands for First, Outer, Inner, Last. This means you multiply the first terms of the binomials, then the outer terms of the binomials (the first term of the first binomial multiplied by the last term of the second), then the inner terms (the reverse of "Outer"), then the last terms in both polynomials. What it gives you, if done correctly, is a trinomial in quadratic standard form.

Example:
(x+3)(x+6)
(x*x)+(x*6)+(3*x)+(6*3)
x2+9x+18

Here's a harder example:
{4x-2)(3x+4)
(4x*3x)+(4x*4)+(-2*3x)+(-2*4)
12x2+16x-6x-8
12x2+10x-8

Simple, eh?

The Foil is one of The three weapons used in the sport of fencing (the others being épée and sabre). The most popular of the weapons, foil is what most fencers learn first.

The foil has a flexible rectangular blade, approximately 35 inches in length, weighing less than one pound. It has a small bell which barely covers the hand of the fencer. Points are scored with the tip of the blade as it is a point-thrusting weapon.

The valid target area in foil is the torso, from the shoulders to the groin, front and back. It does not include the arms, neck, head and legs. The foil fencer's uniform includes a metallic vest (called a lamé) which covers the valid target area, so that a valid touch will register on the scoring machine. A small, spring-loaded tip is attached to the point of the foil and is connected to a wire inside the blade. The fencer wears a body cord inside his uniform which connects the foil to a reel wire, connected to the scoring machine. when the tip is depressed it completes a circuit, if the fencers blade also completes a circuit with his oponents lamé the valid touch light is lit. There are two scoring lights on the machine for each fencer. There is a valid touch indicator which is green for one fencer and red for the other and an "off target" light for each which is white. These "off target" hits do not count in scoring, but they do stop the fencing action temporarily.

In order to score a touch with the foil you need to have "attack priority" which means that you have either initiated the action or parried your opponents initiation and counter-attacked. Actions are initiated by the first fencer to extend his arm towards his opponent. Once an incoming attack is parried it only counts if the counter-attack is missed as the priority has shifted to the attacker's opponent. If both fencers intitiate an action simultaneously then no points are scored.

An interesting technique used by experts with the foil is to whip the blade such that it bends and the point makes contact even thought you are basically making a slashing motion, this is dificult to parry as it happens extremely quickly and it can hit you in some unexpected areas (such as your back while you are facing your opponent).

Also used to refer to wing, as in airfoil or hydrofoil(if the wing is moving through water). In boats you can talk about the foils of a boat, reffering to its keel, centreboard and rudder.

In literature or fiction in general, a foil is character who draws attention to the distinctive qualities of another character (usually the protagonist) through contrast. Often, the foil will be completely opposite the protagonist or exhibit a protagonist's quality to a lesser extent. Sidekicks are the most common embodiment of foils, although rivals, romantic interests, and minor static characters also perform this role.

There are many famous examples of foils:

• Batman and Robin - The Boy Wonder is prime example of a sidekick foil. Robin usually ends up needing help (like getting lowered into a vat of acid), which gives Batman a prompt to look heroic (like swinging across a pit of spikes and laser beams and saving Robin at the last second). Also, Robin's usual ignorance of the bad guy plans highlights Batman's insight of seeing right through them.
• Sherlock Holmes and Dr. Watson - Poor Dr. Watson is another example of a foil. In his confused stupidity, he gives Mr. Holmes every opportunity to show off his intelligence more so than he usually would. Watson asks a stupid question and Holmes replies with "Elementary, my dear Watson," then goes on to explain the plot. Watson's foil role makes Holmes look that much smarter .
• The Fool and King Lear - In Shakespeare's play, King Lear, madness plays an important role. The Fool, a jester type character, acts like he's insane by default by speaking in riddles and barely understandable phrases. However, his craziness doesn't compare to King Lear's disturbed ravings later in the play. This is an example of a foil highlighting a negative characteristic in a protagonist.
• Darth Vader and The Emperor - This famous evil duo from Star Wars shows a different type of foil. It's obvious both Vader and the Emperor are evil, but to emphasize just how evil the Emperor is, Vader takes advantage of an opportunity to redeem himself near the end of the Return of the Jedi. The Emperor shows that Vader wasn't all bad; he had potential for righteousness.
• The killer and the virgin girl that survives - This is more a general theme of foils. In modern slasher movies (Halloween and Friday the 13th), the fact that the killer slices up a bunch of teens having sex while the virgin girl survives draws attention to her purity. By remaining chaste, she is rewarded by not being maimed by the sequel spawning psycho.

Source: Murfin, Ross. The Bedford Glossary of Critical and Literary Terms. Boston, MA: Beford, 1998.

Until the early part of the twentieth century, the only way to apply gold to a book cover was to use gold leaf. It is a painstaking and labor-intensive process. The invention of gold foil in 1932 changed all that.

### The Good News

Gold foil was easy to work with and produced reliable results. For the first time, gold tooling could be mechanised. It revolutionised the titling of mass-market books, and has never been superseded. If you have a book with gold-colored lettering on your shelves right now, it was done using foil.

Although foil is easier to work with than leaf metal, it produces a less detailed result, and is rarely used in fine binding.

### The Structure

Foil is made up of three very thin layers bonded together. It's much thicker than gold leaf on its own, and much easier to handle as a result.

```----------------------  <- plastic membrane
----------------------  <- metal leaf
```
• Plastic Membrane

The plastic membrane gives the foil added tensile strength and cohesiveness. Without it, foil would tear and crumple almost as easily as gold leaf.

Because the membrane doesn't stick to the book during gold tooling, it is possible to design your tooled effect directly on it. It's best to use a stamp pad with water-soluble ink and a felt tip pen. That way, you can wipe off any mistakes with a damp cloth.

• Metal leaf

Originally, foil was made using real gold leaf. It was an obvious innovation to use other, less expensive metals such as aluminum and Dutch metal, as well as non-metallic colorants.

The real innovation, the one that made foil revolutionary, was the adhesive used. Foil adhesive is a bit more sophisticated than glaire. It responds to a wider range of factors, namely:

Some kinds of foil are purely pressure-activated. These are used on heat-sensitive surfaces. Other, heat-activated foils are used in hot stamping processes as well as bookbinding. They respond to a variety of conditions, from very brief, very hot pressure to a long dwell with a relatively cool tool.

Foil adhesive will also stick to leathers treated with acrylic resin, which rejects glaire.

Foil is available from bookbinding and hot stamping suppliers.

Foil (foil), v. t. [imp. & p. p. Foiled (foild); p. pr. & vb. n. Foiling.] [F. fouler to tread or trample under one's feet, to press, oppress. See Full, v. t.]

1.

To tread under foot; to trample.

King Richard . . . caused the ensigns of Leopold to be pulled down and foiled under foot. Knoless.

Whom he did all to pieces breake and foyle, In filthy durt, and left so in the loathely soyle. Spenser.

2.

To render (an effort or attempt) vain or nugatory; to baffle; to outwit; to balk; to frustrate; to defeat.

And by mortal man at length am foiled. Dryden.

Her long locks that foil the painter's power. Byron.

3.

To blunt; to dull; to spoil; as, to foil the scent in chase.

Foil, v. t. [See 6th File.]

To defile; to soil.

[Obs.]

Foil, n.

1.

Failure of success when on the point of attainment; defeat; frustration; miscarriage.

Milton.

Nor e'er was fate so near a foil. Dryden.

2.

A blunt weapon used in fencing, resembling a smallsword in the main, but usually lighter and having a button at the point.

Blunt as the fencer's foils, which hit, but hurt not. Shak.

?socrates contended with a foil against Demosthenes with a word. Mitford.

3.

The track or trail of an animal.

To run a foil,to lead astray; to puzzle; -- alluding to the habits of some animals of running back over the same track to mislead their pursuers.

Brewer.

Foil, n. [OE. foil leaf, OF. foil, fuil, fueil, foille, fueille, F. feuille, fr. L. folium, pl. folia; akin to Gr. , and perh. to E. blade. Cf. Foliage, Folio.]

1.

A leaf or very thin sheet of metal; as, brass foil; tin foil; gold foil.

2. Jewelry

A thin leaf of sheet copper silvered and burnished, and afterwards coated with transparent colors mixed with isinglass; -- employed by jewelers to give color or brilliancy to pastes and inferior stones.

Ure.

3.

Anything that serves by contrast of color or quality to adorn or set off another thing to advantage.

As she a black silk cap on him began To set, for foil of his milk-white to serve. Sir P. Sidney.

Hector has a foil to set him off. Broome.

4.

A thin coat of tin, with quicksilver, laid on the back of a looking-glass, to cause reflection.

5. Arch.

The space between the cusps in Gothic architecture; a rounded or leaflike ornament, in windows, niches, etc. A group of foils is called trefoil, quatrefoil, quinquefoil, etc., according to the number of arcs of which it is composed.

Foil stone, an imitation of a jewel or precious stone.