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Help with Algebra: Learn the FOIL Method for Multiplying Binomials

written by: Joli Ballew • edited by: Elizabeth Stannard Gromisch • updated: 1/4/2012

The FOIL method is an old standard for multiplying two binomials. You use FOIL to multiply binomials similar to this: (3x+2)(4x-1). This will take you through the method, and then provide you with sample questions, and answers, you can practice with.

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    The FOIL Method

    The FOIL method is an old standard for multiplying two binomials. You use FOIL to multiply binomials similar to this: (3x+2)(4x-1). The product of multiplying these particular binomials is 12x² + 5x -2. There are special cases that exist when multiplying binomials though, for instance, when the two binomials produce a difference of squares. (x-3)(x+3) offers up x²-9. In this case, there’s no middle term.

    FOIL stands for First, Outer, Inner, Last. This acronym represents the order in which you should multiply the binomials’ terms to get the product. Multiplying the terms in this order not only keeps the answer in the correct arrangement, but also allows the acronym to read FOIL, instead of something more painful, like LOIF or FLOI.

    The following list details what FOIL stands for and how it’s used in more precise terms:

    · F. F stands the first term of both binomials. Here, the first terms are (3x+2)(4x-1). In this example, you’d multiply 3x and 4x to get 12x².

    · O. O stands for the outer term of both binomials. Here, the outer terms are (3x+2)(4x-1). In this example, you’d multiple 3x and -1 to get -3x.

    · I. I stands for the inner terms of both binomials. Here, the inner terms are (3x+2)(4x-1). In this example, you’d multiply 2 and 4x to get 8x.

    · L. L stands for the last terms of both binomials. Here, the last terms are (3x+2)(4x-1). In this example you’d multiply 2 and -1 to get -2.

    Combine all four answers to get 12x²-3x+8x+-2. Combine the two middle terms to complete the problem: 12x²+5x-2. (-3x+8x=5x, and you can drop the + sign before the -2.)

    There are a few ways to think about FOILing binomials. If you’re the left-brained type, you can label them with letters before multiplying; if you’re the right-brained type you can draw a picture. If you prefer a more hands-on approach, draw the picture and trace the outline with your finger. (You can even move your lips if you want!)

    Here’s an example of how you could label terms as you work.

    F O F O

    (3x+2)(4x-1)

    L I I L

    This will help you stay focused on the task of FOILing.

    You can draw a smiley face to help you remember how to FOIL. Most people prefer this method, as it gives them a visual of what they need to multiply, and the order they need to do it.

    You can draw arcs from the first to first terms and outer to outer terms over the top of the expression and draw arcs from the inner to inner terms and last to last terms on the bottom. When you do, you create a smiley face! You can trace this with your finger if you prefer a more tactile method of solving the problem.

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    Practice Questions

    1.(2x-3)(x+4)=

    2.(7x-3)(4x-1)=

    3.(3x-9)(3x+9)=

    4.(9x+70)(7x-60)=

    5.(2x-3)(2x-3)=

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    Solutions

    1.(2x-3)(x+4) =

    2x(x) + 2x(4)+-3(x)+-3(4)=

    2x²+8x+-3x+-12=

    2x²+5x-12.

    The important thing to watch in this problem is the negative sign before the three. You can add additional parenthesis if you have problems keeping negative signs in check. For instance, consider writing the second line like this: 2x(x) + 2x(4)+(-3)(x)+(-3)(4) if you ended up with the wrong answer due to a missed negative sign. You can also add the extra + signs, as shown in this answer.

    2.(7x-3)(4x-1)=

    7x(4x)+7x(-1)+-3(4x)+-3(-1)=

    28x²+-7x+-12x+3=

    28x²+-19x+3=

    28x²-19x+3.

    If you’re having trouble remembering the rules for multiplying positives and negative numbers, refer to page xx in Chapter xx. If you’re having trouble remembering the rules for adding and subtracting like terms, refer to page xx in Chapter xx. (You can also skip the step of adding the + signs before negative numbers. A negative sign means subtract, and takes the place of the + sign.)

    3.(3x-9)(3x+9)=

    3x(3x)+3x(9)+-9(3x)+-9(9)=

    9x²+27x+-27x+-81+

    9x²+0+-81=

    9x²-81.

    You should have known that the middle terms would cancel out before working the problem. The two binomials have only one difference: One binomial has a subtraction sign and the other an addition sign. Learning this now will make factoring these equations much easier later.

    4.(9x+70)(7x-60)=

    9x(7x)+9x(-60)+70(7x)+70(-60)=

    63x²+-540x+490x+-420=

    63x²+-50x+-420=

    63x²-50x-420.

    Don’t let the big numbers bog you down. They’re just bigger, that’s all, but the procedure is still the same. Hopefully, you’ve learned to skip the fourth step here too; you should be able to jump from 63x²+-540x+490x+-420 to 63x²-50x-420 without having to deal with the extra + signs.

    5.(2x-3)(2x-3)=

    2x(2x)+2x(-3)+ -3(2x)+-3(-3)=

    4x²+-6x+-6x+9=

    4x²-12x+9.

    Notice that the two terms that you combined here were alike. Also notice the two binomials are the same. This will always happen, and understanding that will make factoring later much simpler.

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    I hope the explanation of the FOIL method, and the practice questions and answers have helped you understand this tried and true method for solving binomials in algebra.

    Have some questions of your own? Leave a comment!