Key takeaway: The FOIL method multiplies two binomials by adding four products in order: First, Outer, Inner and Last, then combining the like terms.
Grade 9 – Algebra 1A-APR.A.1Expressions and Equations
Definition of FOIL Method
The FOIL method is a memory trick for multiplying two binomials. It makes sure that every term in the first binomial is multiplied by every term in the second one, which is the distributive property used twice.
FOIL stands for First, Outer, Inner, Last. Multiply the first terms of each binomial, then the outer terms, then the inner terms, and finally the last terms. Four products appear. The outer and inner products usually have the same variable part, so they combine into the middle term of a trinomial. FOIL only fits two binomials; bigger products need the full distributive property. The letters give a fixed order, which keeps students from forgetting one of the four products.
Formula and notation
The FOIL method applied to \((a + b)(c + d)\) gives:
\[(a + b)(c + d) = \underbrace{ac}_{F} + \underbrace{ad}_{O} + \underbrace{bc}_{I} + \underbrace{bd}_{L}\]
Keep signs attached to the numbers. Subtraction inside a binomial is addition of a negative term.
Key facts
- FOIL gives four products, which often shrink to three terms after combining.
- The middle term of the answer comes from adding the outer and inner products.
- Check the work by substituting a simple number, such as x = 1, in both forms.
- Reading a FOIL result backward is how you factor a trinomial into two binomials.
FOIL Method example
Multiply \((3x + 5)(2x – 7)\) using FOIL.
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First: \(3x \cdot 2x = 6x^{2}\). Outer: \(3x \cdot (-7) = -21x\).
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Inner: \(5 \cdot 2x = 10x\). Last: \(5 \cdot (-7) = -35\).
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Add and combine like terms: \(6x^{2} – 21x + 10x – 35 = 6x^{2} – 11x – 35\).
The product is \(6x^{2} – 11x – 35\).
Common mistake
Watch out
Hector multiplies only the first terms and the last terms: \((3x + 5)(2x – 7) = 6x^{2} – 35\).
Why it is wrong
He skipped the outer and inner products, so two of the four required multiplications are missing. Test with \(x = 1\): the original gives \(8 \times (-5) = -40\), his answer gives \(-29\), but the correct \(6 – 11 – 35 = -40\).
Related terms and lessons
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