Product of two binomials
Expand a product by multiplying every term in one binomial by every term in the other.
Product of two binomials is one of 3 polynomials formulas in the algebra section of this library, and it is used at middle school · high school level.
Why product of two binomials works
This is the distributive property used twice. Each term in the first bracket has to meet each term in the second, and two terms meeting two terms makes four pairings. It matches the area of a rectangle chopped into two strips each way, giving four smaller rectangles.
What each symbol means
$a,b,c,d$ may be numbers, variables, or expressions.
Product of two binomials: when it holds
Combine like terms only after all four products are written.
When it stops applying
The four-product count only applies to two brackets of two terms. Multiplying (x + 2) by a three-term expression needs six products, so the popular FOIL mnemonic runs out of letters and quietly leaves terms behind.
Product of two binomials: a worked example
$(x+2)(x-3)=x^2-x-6$.
The mistake to avoid
What people do: Multiplying only the first pair and the last pair, turning (x + 2)(x - 3) into x squared - 6.
Why it goes wrong: The two cross products are missing, so the middle term vanishes. The correct expansion is x squared - x - 6, which differs from the shortcut at almost every value of x.
Do this instead: Draw four arrows before you multiply, or use a two-by-two box, so that no pairing can be skipped.
Product of two binomials: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a,b,c,d$ may be numbers, variables, or expressions.
- Check the conditions before substituting. Combine like terms only after all four products are written.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Polynomials
- Level
- Middle school · High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where product of two binomials comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Factoring Polynomials — the lesson behind this formula: reverse multiplication to expose roots and structure.
- Standard Form Polynomial Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about product of two binomials
Is FOIL the same rule?
Yes, for two binomials it is just a name for the order of the four products. It stops being useful the moment either bracket has more than two terms, while distributing always works.
What happens when both brackets are identical?
You get a perfect square with a doubled middle term, because the two cross products are the same. Squaring x + 5 gives x squared + 10x + 25, not x squared + 25.
Do I always end up with four terms?
No. Like terms usually combine into three, and when the cross products are exact opposites they cancel to two, which is how the difference of squares pattern appears.
How can I check an expansion quickly?
Substitute a simple number into both forms. At x = 1, the brackets (1 + 2)(1 - 3) give -6, and x squared - x - 6 also gives -6, so the expansion survives the test.