FOIL stands for "first, outer, inner, and last." It refers to the corresponding variables and numeric expressions in a polynomial. The letters F-O-I-L lets the math student recall what order multiplication occurs.
An Example of FOIL of a Polynomial
(x-1)(x+6)= x^2+5x-6
There are two factors in this equation. The first factor is (x-1), while the second factor is (x+6). The two factors are determined as follows:
Step 1: What multiplied by what could result in x^2?
The easiest thing would be x times x.
Step 2: What multiplied together could result in a positive 5 x when the coefficients are added together?
The list could include (x, 4x), (2x, 3x), (3x, 2x), (4x, x), but also could include a negative answer (-x, 6x), since -1 plus 6 also equals 5.
Step 3: See how there are multiple possibilities, especially when signs are switched? There is only one correct way though! This is why the FOIL process helps greatly when working with polynomials/quadratic equations. The correct combination for getting 5x in this case is (-x, 6x), since the final step of factoring a polynomial is the multiplication of the last term in each factor to form the final numeric variable, which in this case, is -6. Since -6 is negative, there must be one positive and one negative variable in Step 2.
The Process of F-O-I-L
Step 1: multiply first terms
In this case, multiply x by x.
Step 2: multiply outer terms
In this case, multiply the x from the first factor by 6 from the second factor.
These are the outermost variables in the polynomial.
Step 3: multiply inner terms
These are the two variables which are closest to each other.
In this case, multiply -1 by x (with x coming from the second factor of x+6).
Step 4: multiply last terms
These are the last values in each factor.
In the sample problem, these last terms are -1 and 6.
Using FOIL Method to Find a Polynomial
If one follows steps 1-4 using the example (x-1)(x+6), it will look like this:
(x*x) [first] + (x*6) [outer] + (-1*x) [inner] + (-1*6)
Now, one may simplify by multiplying what is inside each set of parentheses.
The next step would look like:
x^2+6x-x-6
There is a set of like terms, both of which have an x in them. These like terms can be combined.
Therefore, now the expression is:
x^2+5x-6
This expression should look familiar, since it is the original polynomial expression in its non-factored form.
Polynomials can be factored into expressions that make it easier to determine the numeric value of a variable. The process used is called FOIL. It uses the "first, outer, inner, and last" values for multiplication in order.
 |
