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Binomial

Binomial

A polynomial with two terms which are not like terms. The following are all binomials: 2x – 3, 3x5 +8x4, and 2ab – 6a2b5.

 

 

See also

Monomial, trinomial

Worked Example

Problem: Identify which of the following expressions are binomials: (a) 5x2+3x5x^2 + 3x, (b) 7y−2y+47y - 2y + 4, (c) 9m39m^3, (d) x2−16x^2 - 16.
Step 1: Check expression (a): 5x2+3x5x^2 + 3x has two terms, 5x25x^2 and 3x3x. These are not like terms because the exponents on xx differ. This is a binomial.
5x2+3x✓5x^2 + 3x \quad \checkmark
Step 2: Check expression (b): 7y−2y+47y - 2y + 4 appears to have three terms, but 7y7y and −2y-2y are like terms. Combine them first: 7y−2y=5y7y - 2y = 5y. The simplified form is 5y+45y + 4, which has two unlike terms. This is a binomial.
7y−2y+4=5y+4✓7y - 2y + 4 = 5y + 4 \quad \checkmark
Step 3: Check expression (c): 9m39m^3 has only one term. A single-term expression is a monomial, not a binomial.
9m3(monomial, not a binomial)9m^3 \quad \text{(monomial, not a binomial)}
Step 4: Check expression (d): x2−16x^2 - 16 has two terms, x2x^2 and −16-16. These are not like terms. This is a binomial.
x2−16✓x^2 - 16 \quad \checkmark
Answer: Expressions (a), (b) after simplifying, and (d) are binomials. Expression (c) is a monomial.

Another Example

Problem: Multiply the binomials (x+3)(x−5)(x + 3)(x - 5) using the FOIL method.
First: Multiply the first terms of each binomial.
x⋅x=x2x \cdot x = x^2
Outer: Multiply the outer terms.
x⋅(−5)=−5xx \cdot (-5) = -5x
Inner: Multiply the inner terms.
3⋅x=3x3 \cdot x = 3x
Last: Multiply the last terms of each binomial.
3⋅(−5)=−153 \cdot (-5) = -15
Combine: Add all four products and combine like terms.
x2−5x+3x−15=x2−2x−15x^2 - 5x + 3x - 15 = x^2 - 2x - 15
Answer: (x+3)(x−5)=x2−2x−15(x + 3)(x - 5) = x^2 - 2x - 15

Frequently Asked Questions

Is x+2xx + 2x a binomial?
No. Although x+2xx + 2x appears to have two terms, those terms are like terms. When you combine them you get 3x3x, which is a single term — a monomial. A binomial must have exactly two unlike terms after simplification.
Can a binomial have more than one variable?
Yes. An expression like 3xy−4z23xy - 4z^2 is a binomial because it has exactly two terms that are not like terms. The number of distinct variables does not matter; what matters is having exactly two unlike terms.

Binomial vs. Trinomial

A binomial has exactly two unlike terms (e.g., x+7x + 7), while a trinomial has exactly three unlike terms (e.g., x2+3x+7x^2 + 3x + 7). Both are specific types of polynomials, classified by their number of terms. A monomial, by contrast, has just one term.

Why It Matters

Binomials appear throughout algebra and beyond. Multiplying two binomials is one of the most common operations in algebra, and the FOIL method is designed specifically for it. Many important patterns — the difference of squares a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b), perfect square trinomials, and the binomial theorem used in probability and combinatorics — all revolve around binomials.

Common Mistakes

Mistake: Counting terms before combining like terms, so expressions like 4x+2x−14x + 2x - 1 are mistakenly called trinomials.
Correction: Always simplify first. 4x+2x−1=6x−14x + 2x - 1 = 6x - 1, which has two unlike terms and is therefore a binomial.
Mistake: Thinking the two terms of a binomial must be separated by a plus sign. Students sometimes believe x2−9x^2 - 9 is not a binomial because it uses subtraction.
Correction: Subtraction is just addition of a negative term. The expression x2−9x^2 - 9 has two unlike terms, so it is a binomial regardless of the sign.

Related Terms

  • Polynomial — General category that includes binomials
  • Monomial — Polynomial with exactly one term
  • Trinomial — Polynomial with exactly three terms
  • Term — Building block counted in a binomial
  • Like Terms — Must be combined before counting terms
  • FOIL — Method for multiplying two binomials
  • Difference of Squares — Special binomial factoring pattern