Multiply Binomials Using the FOIL Method

When multiplying two binomials you must use the distributive property to ensure that each term is multiplied by every other term. This can sometimes be a confusing process, as it is easy to lose track of which terms you have already multiplied together. You can use FOIL to multiply binomials using the distributive property in an organized way.[1]By simply remembering the words in the acronym, this method will help you multiply binomials quickly.

Steps

Setting Up the Problem

  1. Write the two binomials side-by-side in parentheses. This setup helps you easily keep track of operations when using the foil method.
    • For example, if you are multiplying <math>2x - 7</math> and <math>5x + 3</math>, you would set up the problem like this:
      <math>(2x - 7)(5x + 3)</math>
  2. Ensure you are multiplying two binomials. A binomial is an algebraic expression with two terms.[2] The FOIL method does not work when multiplying trinomials, or a binomial by a trinomial.
    • A term is a single number or variable, such as <math>3</math> or <math>x</math>, or it could be a multiplied number and variable, such as <math>3x</math>.[3]
    • Read Multiply Polynomials for instructions on multiplying other types of polynomials.
    • For example, you could NOT multiply <math>(2x - 4)(3x^{2} - 2x + 8)</math> using the FOIL method, because the second expression is a trinomial, with three terms.
    • You could multiply <math>(2x - 7)(5x + 3)</math>, because both expressions are binomials, with two terms each.
  3. Arrange the binomials by terms. Most algebra problems will already be arranged this way, but if not, make sure the first term in each expression contains the variable, and the second term in each expression contains the coefficient.
    • Setting up the problem this way makes simplifying easier.
    • A coefficient is a number without a variable.
    • For example, you would change <math>(2x - 7)(3 + 5x)</math> to <math>(2x - 7)(5x + 3)</math>.

Multiplying Binomials

  1. Multiply the first terms in each expression. The F in FOIL stands for “first.”
    • Remember when multiplying a variable by itself, such as <math>x \times x</math>, the result is a squared variable (<math>x^{2}</math>).
    • For example, if your problem is <math>(2x - 7)(5x + 3)</math>, you would first calculate:
      <math>(2x)(5x)</math>
      <math>= 10x^{2}</math>
  2. Multiply the outside terms in each expression. The O in FOIL stands for “outside,” or “outer.” The outside terms are the first term of the first expression, and the last term of the second expression.
    • Pay close attention to addition and subtraction. If the second binomial is a subtraction expression, that means in this step you will be multiplying a negative number.
    • For example, for the problem <math>(2x - 7)(5x + 3)</math>, you would next calculate:
      <math>(2x)(3)</math>
      <math>= 6x</math>
  3. Multiply the inside terms in each expression. The I in FOIL stands for “inside,” or “inner.” The inner terms are the last term of the first expression, and the first term of the second expression.
    • Pay close attention to addition and subtraction. If the first binomial is a subtraction expressions, that means in this step you will be multiplying a negative number.
    • For example, for the problem <math>(2x - 7)(5x + 3)</math>, you would next calculate:
      <math>(-7)(5x)</math>
      <math>= -35x</math>
  4. Multiply the last terms in each expression. The L in FOIL stands for “last.”
    • Pay close attention to addition and subtraction. If either binomial is a subtraction expression, that means in this step you will be multiplying a negative number.
    • For example, for the problem <math>(2x - 7)(5x + 3)</math>, you would next calculate:
      <math>(-7)(3)</math>
      <math>= -21</math>
  5. Write the new expression. To do this, write out the new terms you created during the FOIL process. You should have four new terms.
    • For example, after multiplying <math>(2x - 7)(5x + 3)</math>, your new expression is <math>10x^{2} + 6x - 35x - 21</math>.
  6. Simplify the expression. To do this, combine like terms. Usually you will have two terms with the <math>x</math> variable that need to be combined.
    • Pay close attention to positive and negative signs as you add or subtract.
    • For example, if your expression is <math>10x^{2} + 6x - 35x - 21</math>, you would simplify by combining <math>6x - 35x</math>. Thus, the expression simplifies to <math>10x^{2} - 29x - 21</math>



Tips

  • You can think of this as two separate distributions: (2x)(5x + 3) added with (-7)(5x + 3)

Things You'll Need

  • Paper
  • Pencil or pen
  • To know how to multiply, add, and subtract

Related Articles

Sources and Citations

  1. ↑ http://www.coolmath.com/prealgebra/15-intro-to-polynomials/06-polynomials-multiplying-foil-01-108
  2. ↑ https://www.mathsisfun.com/definitions/binomial.html
  3. ↑ https://www.mathsisfun.com/definitions/term.html
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