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How Do You FOIL Multiply Polynomials?

Published in Polynomial Multiplication 3 mins read

The FOIL method is a specific way to multiply two binomials, ensuring every term in the first binomial is multiplied by every term in the second binomial. It's an acronym that stands for First, Outer, Inner, Last.

Understanding the FOIL Method

FOIL is a mnemonic used to remember the steps needed to multiply two binomials systematically. Each letter represents a pair of terms to multiply:

  • First: Multiply the first terms of each binomial.
  • Outer: Multiply the outer terms of the two binomials.
  • Inner: Multiply the inner terms of the two binomials.
  • Last: Multiply the last terms of each binomial.

After performing these four multiplications, you typically combine any like terms to simplify the result.

Step-by-Step FOIL Multiplication

Let's use the example implied by the reference to illustrate the FOIL method: multiply \((x + 3)(x - 2)\).

Here's how you apply each step of FOIL:

  1. First: Multiply the first term in the first binomial by the first term in the second binomial.
    \(x \times x = x^2\)
  2. Outer: Multiply the outer term in the first binomial by the outer term in the second binomial.
    As mentioned in the reference: "Then we do the outer terms x times negative 2 or negative 2x."
    \(x \times (-2) = -2x\)
  3. Inner: Multiply the inner term in the first binomial by the inner term in the second binomial.
    As mentioned in the reference: "Then the inner terms 3 times X or 3x."
    \(3 \times x = 3x\)
  4. Last: Multiply the last term in the first binomial by the last term in the second binomial.
    \(3 \times (-2) = -6\)

Combining Terms

After completing the four multiplications using the FOIL steps, you add the results together:

\(x^2 + (-2x) + 3x + (-6)\)
\(x^2 - 2x + 3x - 6\)

Now, combine the like terms (the terms with 'x'):

\(-2x + 3x = 1x\) or \(x\)

So, the simplified product is:

\(x^2 + x - 6\)

Summary of FOIL Steps

Here's a quick breakdown:

  • Multiply First terms
  • Multiply Outer terms
  • Multiply Inner terms
  • Multiply Last terms
  • Add the four products
  • Combine like terms

This systematic approach ensures that each term from the first binomial is multiplied by each term from the second binomial, covering all combinations.

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