Q: What is multiplying binomials used for?

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It's the difference between multiplication and division. Multiplying binomials is combining them. Factoring polynomials is breaking them apart.

Combining like terms.

When multiplying numbers with exponents, you add the exponents.

binomials

Foil

multiplying

It is only not needed if you know of another method. If FOIL is the only way you know to multiply two binomials, then it is definitely needed.

You don't need any acronym; just multiply every monomial on the left with every monomial on the right. The same goes for multiplying a binomial with a trinomial, two trinomials, or in fact for multiplying any polynomial by any other polynomial.

The same way that factoring a number is different from multiplying two factors. In general, it is much easier to multiply two factors together, than to find factors that give a certain product.

Depends on the kind of binomials. Case 1: If both binomials have different terms, then use the distribution property. Each term of one binomial will multiply both terms of the other binomial. After distribution, combine like terms, and it's done. Case 2: If both binomials have exactly the same terms, then work as in the 1st case, or use the formula for suaring a binomial, (a ± b)2 = a2 ± 2ab + b2. Case 3: If both binomials have terms that only differ in sign, then work as in the 1st case, or use the formula for the sum and the difference of the two terms, (a - b)(a + b) = a2 - b2.

does the FOIL system work for any binomials

The advantage of recognizing some special binomials is that the math can then be done much more quickly. Some of the binomials appear very frequently.