Introduction: How to FOIL

FOIL is an acronym used in mathematics which stands for Firsts Outers Inners Lasts. It is a process used to multiply polynomials of two terms together. A polynomial is a sum of terms, and a term is just a constant (any number) multiplied by a variable (some unknown number). An example of a polynomial would be 5x + 2, where 5x and 2 are the terms of the polynomial. FOILing is a fundamental mathematical skill used throughout algebra, calculus, and other high level math courses.

In order to complete this process, you should be able to add multiply.

You will need:

  • Paper
  • Pencil
  • Two polynomials consisting of two terms to multiply

It typically takes around 3 minutes to complete this process.

In this example we will multiply together two simple polynomials, x+1 and x+2.

Step 1: Firsts

Start by multiplying the first term of each polynomial together. In this case the first term for each is x, so we will multiply x times x.

Step 2: Outers

Next, we will multiply the terms on the outside of the two polynomials. In this case, an x and a 2 are on the outside, so we multiply x times 2.

Step 3: Inners

Now we will multiply the terms on the inside (1 and x).

Step 4: Lasts

Lastly, we will multiply the last term of each polynomial together to obtain 1 times 2 equals 2.

Step 5: Add Terms Together

Add the terms we obtained together to get our answer.

Step 6: Simplify

Now we want to add like terms together to obtain the simplest answer possible. In this case we will add 2x and x together to obtain 3x.

Note:

We were able to add 2x and x (notice: x = 1*x) together because both terms are a number multiplied by x. We could not have added x and 3 together because 3 does not have an x attached to it.

Step 7: Conclusion

You have now successfully multiplied two polynomials together to obtain a new polynomial by FOILing.

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