However, in contrast to the FOIL method, which is 100% reliable and never fails, the reverse FOIL method works by trial-and-error: we try to guess the First and Last terms of the two linear binomials based on the values of ax² and c so that the sum of the Outer and Inner products equals bx. In particular, the reverse FOIL method aims at factoring a quadratic trinomial, i.e., a polynomial of the form ax² + bx + c, into the product of two linear binomials. The reverse process would be to convert the sum of some monomials into the product of two binomials. In other words, it takes two binomials and returns the sum of four monomials (or fewer than four, if simplifications are possible). Simplifying 7x⁴ + 2x⁴ to 9x⁴, we arrive at the final answer:Īs we've explained above, the FOIL method helps you to multiply two binomials. No simplification is possible, so the final answer is: We can combine the middle two terms, -4x + 6x = 2x, so the final answer is: Remember that the FOIL method only works for multiplying binomials! For more general polynomials, you need to use other methods.Īs the best way to learn the FOIL method is to go through a few examples, let's go ahead and get our hands dirty with three examples: (The rule stated above may also be remembered by the word FOIL, suggested by the first letters of the words first, outer, inner, last.) first terms, outer terms, inner terms, last terms. He intended it as a mnemonic for students. It was William Betz who came up with the word FOIL in his 1929 text Algebra for Today. Obviously, the order in which the four terms appear in the sum does not matter and does not need to match the order F-O-I-L.

We need to add the four FOIL terms together. When we've calculated the four partial products ( First, Outer, Inner, and Last), there is one last step before we reach the final answer. That is, the second term of the first binomial and the second term of the second binomial. That is, the second term of the first binomial and the first term of the second binomial. I ➺ Inner terms are multiplied (the middle two terms of the binomials). That is, the first term of the first binomial and the second term of the second binomial. O ➺ Outer terms are multiplied (the terms located at the outermost ends of the binomials). That is, the first term of the first binomial and the first term of the second binomial. The strange word FOIL is an acronym for the four terms appearing in the final result.į ➺ First terms of each binomial are multiplied. In other words, the FOIL method allows you to go from The FOIL method allows you to multiply two binomials quickly, that is, without performing all the intermediate steps. This is when the FOIL method enters the stage. Note that in total, we've applied the distributive property thrice. (ax + b) * (cx + d) = acx² + adx + bcx + bd Each of the two resulting terms gets further distributed:

That is, the binomial cx + d is distributed over the addition in the binomial ax + b. Suppose now that we want to multiply two linear binomials ax + b and cx + d. Of the four binomials listed above, only the first two, i.e., 3 + x and 3x + 3 are linear binomials. 3 + x, 3x + 3, x³ + 1, 4x³ + x² are binomials.Ī binomial whose highest degree is equal to one is called a linear binomial.Let's recall that a binomial is the sum of two monomials, and a monomial is a product of a number (coefficient) and variable (unknown) raised to a non-negative integer power (neither negative powers nor fractional powers are allowed). FOIL is a method for multiplying two binomials.

Foil math calculator