Key takeaway: A binomial is a polynomial with exactly two unlike terms joined by addition or subtraction, such as 3x + 8 or a² − 49.
Grade 9 – Algebra 1A-APR.A.1Expressions and Equations
Definition of Binomial
A binomial is a polynomial with exactly two terms. The expressions \(3x + 8\), \(m^{2} – 49\) and \(5a^{3}b – 2b\) are two-term examples. The prefix “bi” means two. The two terms must be unlike, because \(4x + 9x\) simplifies to the single term \(13x\), which is a monomial.
They are everywhere in Algebra 1. Linear expressions such as the cost \(18 + 9.5h\) have two terms, and so do the factors of many quadratics, such as \((x + 4)\) and \((x – 9)\). A phone plan with a flat fee plus a per-gigabyte charge is a two-term story. Adding or subtracting them means combining like terms; the result can have fewer terms than you started with, or it can still be a binomial.
Formula and notation
The general shape is
\[ax^{m} + bx^{n}\]
where the terms have different variable parts. A sign between the terms stays attached to the term that follows it: \(m^{2} – 49\) means \(m^{2} + (-49)\). Its degree is the larger of the two term degrees.
Key facts
- A binomial has two terms; a monomial has one and a trinomial has three.
- Terms must be unlike, or they merge into one term.
- Raised to a power, like (x + 3)², expands into more terms.
- Adding two of them may leave two terms, one term or three terms.
Binomial example
Subtract: \((5m^{2} + 8) – (2m^{2} – 3)\). How many terms does the result have?
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Distribute the minus sign: \(5m^{2} + 8 – 2m^{2} + 3\).
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Combine the \(m^{2}\) terms: \(5m^{2} – 2m^{2} = 3m^{2}\).
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Combine the constants: \(8 + 3 = 11\).
The difference is \(3m^{2} + 11\), which again has two terms.
Common mistake
Watch out
Pia changes only the first sign when subtracting: \(5m^{2} + 8 – 2m^{2} – 3 = 3m^{2} + 5\).
Why it is wrong
The minus sign must multiply every term inside the second parentheses, so \(-(2m^{2} – 3)\) becomes \(-2m^{2} + 3\). Her \(-3\) should be \(+3\), and the constant is \(8 + 3 = 11\), not 5.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 1 vocabulary · Expressions and Equations